The leading theory of the origins of the Indo-European languages, called the Kurgan hypothesis, formulated in the 1950s by Marija Gimbutas, put the origins of the Indo-European languages on the Pontic-Caspian Steppe around the 4th millenium BCE. The other leading theories put it even further back in history.
The Kurgan hypothesis, named after the burial mound culture found across Europe and Central Asia, associates the rise of the Indo-Europeans with the domestication of the horse around 4000 BCE.
History and archeology suggests instead that the arrival of the Indo-European language is associated with disposal of the dead through cremation and with the spread of iron. Horses were an important part of the Indo-European package, but they did not originate it.
Simply put, there are no historically or archeologically documented events traceable to Indo-European cultures or associated iron or cremation techniques prior to 2200BCE, and this evidence takes a couple of hundred years to firm up. Call it the Cremation hypothesis. But the appearance of Indo-European cultures closely coincides with the appearance of iron and cremation, and with the first known iron making cultures (apart of iron from meteroids) appearing in Indo-European language speaking cultures in the areas where Indo-European languages are spoken. This puts the proto-Indo-Europeans 1500-2000 years later than the Kurgan hypothesis although it doesn't necessarily dispute the geographic origins of the proto-Indo-Europeans very strongly.
Clearly, the Indo-European culture was a dynamic and expanding one. It wouldn't have subsumed almost all of Europe, Afghanistan, Iran and South Asia if it wasn't. But, it seems highly unlikely that such a dynamic and expanding culture would remain static and confined to a small area for 1500 or more years before making its debut on the world scene, and a language spoken 1500 years earlier by a society that probably wasn't literate and lacked a strong state and had undergone a major transformation in religious practices evidence by new ways of disposing of the dead would have been barely recognizeable as the same language. The proto-Indo-European language would have changed in all respects in ways that would make it distant from the language that was shared before the Hittites and Tocharian speakers, Sanskrit speakers, and paleo-Baltic language speakers went their separate ways.
The exact place of orign of the Indo-Europeans isn't entirely clear. It could have been anywhere from the Balkans to Kazakhstan. Linguistic connections to other language groups are obscure and many of the closet relatives of proto-Indo-European, of which the Dardic languages and Hittite languages as they were spoken in the 1700s BCE are probably the most basal known branches, are obscure.
The Germanic, and Italic branches of the language are probably descendants of the Hittite language that probably have roots around 1100-1000 BCE in the post-Bronze Age collapse era, filling a void created by the collapse of Minoan culture, with the Celtic branch probably derived from the Italic branch around 700 BCE. The Hittite people, in turn, became a distinct society around 2000 BCE in central Anatolia and probably originated somewhere to the East of there, and the Tocharian language people headed in the opposite direction from the same source population probably around the same time (or perhaps earlier, as they seem to be the only group of Indo-Europeans who didn't practice cremation or sky burial, at least at first, suggesting that this pratice arose after their split in the Hittite group or was adopted when the Tocharians merged with the local people).
The timing is right for Greek to have diverged from Hittite and founded colonies across the Aegan sea from Anatolia, but both linguistic and historical considerations suggest that Greek may have a closer connection to the Indo-Iranian languages through extinct paleo-Balkan languages to its North.
The Indo-Aryans (Sanskrit speaking people) probably arrive in North Pakistan around 1900 BCE, and probably have a common source with the Hittites no long before then. But, Indo-Aryans and Hittites probably had some sort of trade and communication through at least 1800 BCE through which iron smelting technology was shared. Alternately, the Indo-Aryans are an offshoot of the Hittites, probably through the Caucuses and influenced by the people through whom they passed, with little subsequent communications and evidence of earlier iron working is simply due to a lack of archeological study.
The Mittani leadership is probably a direct Western movement of the Indo-Aryan founding group allowing that group 450 years to make its way West (and a shorter time after the fall of the Harappan empire through which the Mittani leadership probably traveled) but fell to the Hittites from whom there were long separated, around two hundred years later.
The Indo-European Iranian languages probably developed from emigration of Indo-Aryans to the West starting around 1500 BCE from India starting at the North in Yaz and slowing gaining ground Southward through the mountains of Iran.
One of the better ways to illustrate the youth of the Indo-European language group is with a selection of dates.
* 4500 BCE Beginning of megalithic culture in the Atlantic and North Mediterranean.
* 4000 BCE Early Kurgan (burial mound) culture in Eastern Europe and Central Asia.
* 4000 BCE Earliest bronze metallurgy in the Caucuses.
* 3500 BCE Non-Indo-European Sumerian language becomes first written language. Egyptian, also a non-Indo-European language of a different family (it is Afro-Asiatic) appears in writing a few hundred years later.
* 2400 BCE Akkadian empire replaces Sumerian with a Semitic language (from the Afro-Asiatic language family).
* 2200 BCE - 1600 BCE The Sintashta-Petrovka-Arkaim culture in the Southern Urals, northern Kazakhstan is a strong candidate for the Indo-Iranian proto-group. They conducted chariot burials and engaged in copper mining and metallurgy. Ancient DNA from the region in this era suggest that about 90% of both paternal and maternal ancestry was west Eurasian in origin and that "at least 60% of the individuals overall had light hair and blue or green eyes."
* 2100 BCE - 1950 BCE Earliest smelted iron objects found at the Kaman-Kalehoyuk archaeological site in Turkey were there are also indications of iron working, about 100 kilometers southeast of Ankara (between the ancient city of Hattsua and the ancient city of Kadesh near the Red River). This would have been close to the formative area for the Hittites, but could also have been a non-Indo-European Hattite site. This location would not be inconsistent with Kussara, the original city-state of the Hittites.
* 2100 BCE - 1450 BCE Linear A script used by speakers of non-Indo-European Minoan language, which was probably related to the pre-Indo-European Etruscan (Tuscany), Rhaetic (roughly speaking Venice), and Lemnian (a Greek island) languages.
* 2004 BCE Semitic speaking Akkadian empire replaced by Kassite empire from the mountains to the East where a Hurrian language (a non-Indo-European, non-Afro-Asiatic language related to the languages of the North Caucuses) replcae them.
* 2000 BCE Cremation begins to appear in the Pannonian plain and along some locations on the mid-Danube River basin (i.e. generally in the Balkans and coast of the Black Sea and the Sea of Marmara which is between the Black and Aegean seas).
* 2000 BCE Andronovo culture invents spoke-wheeled chariot which is strongly associated with the Indo-Iranians.
* 2000 BCE First Hittite iron working.
* 1900 BCE First example of conversion from burial to cremation in South Asia found in North Pakistan at Cemetary H site.
* 1900 BCE-1500 BCE Approximate era of Indo-Aryan (a.k.a. Vedic) entry into North India. The Indo-Aryans spoke Sanskrit which is the language from which the Indo-European languages of South Asia descend.
* 1800 BCE First example of iron use in North India.
* 1800 BCE Mummies with European paternal genes and physical features found in Tarim basin starting a culture that continues until the 9th century CE when replaced by ethnically (North)East Asian Uygars. The Tarim basin was home to the Indo-European Tocharian languages most closely related to Hittite linguistically.
* 1754 BCE The first king of the Hittites, Pithana, captures the city of Kadesh (previously a Hattite city with an Akkadian merchant colony attached to it) from Kussara, their original single city-state. The site of Kussara is unknown but was also in Anatolia. All Anatolian languages in the Indo-European family have their source in Hittite, and all the lands conquered by the Hittites previously spoke Hatti or Hurrian, both of which are non-Indo-European languages. Hittite is the first Indo-European language attested in writing. "Ancient historians and surviving contemporary documents associate iron-working with the Hittites."
* 1700 BCE - 1100 BCE the Rigveda was composed in the north-western region of the Indian subcontinent.
* 1700 BCE Estimate divergence date of Dardic languages of North Pakistan from the Indo-European languages of India based upon preservation of early Rigvedic Sanskrit features not preserved in the languages of India.
* 1600 BCE No iron objects have been found outside the area bounded by Anatolia, Mesopotamia, Egypt, and Crete before this date (other than in South Asia). In this era, iron is more valuable than gold. The Mesopotamian, Egyptian and Cretian supplies appear to have been obtained through trade with Anatolia. All ancient iron mines were in Anatolia and the Caucuses.
* 1550 BCE- 1460 BCE Sanskrit speaking, Indo-Aryan god worshipping elite establishes itself in Mitanni empire in Northeast Mesopotamia (from 2004 BC until the Mitanni, this was the Kassite empire, which spoke Hurrian, a non-Indo-European language related to the modern languages of the North Caucuses). The Mitanni were literate and were experts on horses. The Mittani came to Mesopotamia from somewhere to the East. Notably, they did not speak the language of the now geographically closer Indo-European language group that includes Persian.
* 1500 BCE - 1300 BCE Fedorovo culture in southern Siberia one of the first to practice cremation.
* 1500 BCE- 1100 BCE Yaz culture near the modern North Iranian town of Gorgan (just East of the Caspian Sea) is "an early Iron Age culture . . .regarded as a likely archaeological reflection of early East Iranian culture as described in the Avesta." There are no signs of burials there indicative of disposal of bodies by "sky burial" (i.e. open exposure) or open cremation.
* 1500 BCE Smelted iron objects appear in mainland Greece.
* 1500 BCE- 1000 BCE The Balto-Slavic dialect ancestral to Proto-Slavic is estimated on archaeological and glottochronological criteria to have formed.
* 1450 BCE The non-Indo-European Kaskians regularly invade Hittite territory from the East coast of the Sea of Maramara and the adjacent Anatolian plains which the Hittites never manage to control.
* 1400 BCE Linear B writing of Mycenaean Greek, an early version of the Indo-European language appears around the time that the Mycenaean Greeks replace the Minoans in Crete. The Myceneans conquered the Minoans from a colony on the Greek mainland (Mycenaea). The Myceneans in turn, according to Greek history have their roots further to the North of mainland Greece.
* 1350 BC Iron objects had found their way to the eastern Danube.
* 1321 BCE Hittite empire has expanded to reach all of Mediterranean coast of Anatolia reaching as far to the South in the Levant as Damascus, the central Anatolian Black Sea coast, and the upper Euphrates Valley in all areas which are part of modern Syria.
* 1274 BCE Hittite empire stopped by Egyptians in major battle of Kadesh in the Levant (even though the Hittites basically win).
* 1200 BCE Bronze age collapse and the demise of the Hittite empire, widespread destruction of major cities in the Greek and coastal Anatolian region, the Sea People coming from areas previously associated with the Myceneans and Hittites attack locations up and down the Eastern Mediterranean. The proximate cause of the demise of the Hittite empire is an invasion of the Kaskians from the West in alliance with the Mushki (possibly a proto-Georgian people) from the East at a time when the Hittite empire had been weakened by a civil war, famine and a string of major earthquakes.
* 1200 BCE End of megalithic culture (associated, for example, with Stonehenge) in Atlantic Europe and the North Mediterranean.
* 1150 BCE The Kaskians are defeated by an Assyrian king and disappear from history. Some may have assimilated into the proto-Georgian population.
* 1100 BCE Beginning of late Nordic Bronze Age in Southern Scandinavia. Cremation begins to appear in Southern Scandinavia and many metal objects related to horses are found.
* 1000 BCE Estimated date of composition of the Avesta, the primary sacred book of the Zorostrian religion, in Iran, by Zarathushtra (Zoroaster) himself in Avestian a member of the Iranian branch of the Indo-European language family.
* 1000 BCE-700 BC Iranian language family language (an Indo-European language) speaking Sythians replace a non-Indo-European language speaking population on the Pontic-Caspian Steppe.
* 1000 BC Sometime after this date various Indo-European paleo-Balkan languages, now all extinct were spoken in the Balkans to the North of Greece and noted by the classical Greeks. These included the Ancient Macedonian, Dacian, Thracian, Illyrian and Paionian languages in different regions of this area, they may have been related but this is not well established. Albanian, which is an Indo-European language, may have its origins from these languages.
* 800 BC Earliest written evidence of the Phrygian language, found near the core of the former Hittite empire but believed to be more closely related to Greek and Armenian and possibly the source language for Armenian. Probably arrived in Anatolia after the fall of the Hittite empire.
* 700s BC Greek colonies founded on the Mediterranean coast in Iberia.
* 753 BC Rome founded by Romulus according to Roman tradition. Rome was initially a small Indo-European language speaking city state dominated by the neighboring non-Indo-European speaking Estrucans of modern day Tuscany until around 500 BC when Rome assimilated the Estrucans into their state. By the 1st century CE Estrucans was a dead language. Ancient DNA evidence indicate the the genetic lineages associated with the Estrucans who received elite burials had vanished within a thousand years of Roman control. There were several small Italic language speaking states in Southern Italy around the same time which were probably founded around the same time and which were soon assimilated into Rome with their languages going extinct.
* 750 BCE Germanic languages start to expand from Scandinavia South. The Germanic languages are show similarities with the side of the Indo-European language family that includes the Celtic, Italic, Anatolian and Tocharian languages rather than with the side that includes the Indo-Iranian, Armenian, Mittani elite, Hellenic and Balto-Slavic languages.
* 700 BCE Hallstatt culture associated with formation of the Celtic languages emerges in Central Europe. There are strong linguistic ties between Celtic and the Indo-European Italiann languages. The Italic and Celtic languages, in turn, show strong similarites with the Anatolian languages (as opposed to the Greek languages).
* 600s BCE Celtic cultures bring iron to Iberia.
* 600 BCE Cremation was common in the Italian pennisula after burial was used for thousands of years before that time.
* 500s BCE to 0 BCE The Indo-European Messapian (also known as Messapic) language is spoken in Southeastern Italy, in the region of Apulia by three Iapygian tribes of the region: the Messapians, the Dauni and the Peucetii. "Messapian may have been one of the Illyrian languages, which were spoken mainly on the other side of the Adriatic Sea. This speculation is based on personal names found on tomb inscriptions and on classical references."
* 219 BCE Romans enter Iberia.
* 476 CE Fall of Western Roman empire. Empire is divided between the Ostrogoths (Italy), VisiGoths (Iberia), and Vandals (North Africa). The Goths migrated from Gotland in the Baltic Sea to central Asia around 100 CE. The fall of the Roman empire leads to the division of Latin into the many Romance languages.
* 600 CE Sudden expansion of Slavic languages which expand until they reach areas where the Uralic languages (e.g. Finnish and Estonian) have already established themselves.
History and archeology, informed by important archeological finds and ancient DNA analysis since the 1950s, favor a much earlier origin for Indo-European languages. Dates in historical documents which can be calibrated with dates from parallel documents reporting the identical events (such as the reigns of kings), relative dating through layers in archeological digs, and carbon dating of material found in archeological digs, together are very accurate (to within decades). This is much more accurate than the dating made possible though linguistic analysis and population genetics, which rely on assumptions about rates of change that have little calibration and are not known to be stable in the relatively short period of time known in ancient history. So, whenever possible, historical and archeological dates should be favored over linquistic ones.