One of the hot fields in population genetics and anthropology in the past decade or so has been the extraction and analysis of DNA evidence from hominin remains. These have included Neanderthal DNA, European hunter-gatherers who are modern humans, and contemporary modern humans.
Neanderthals v. Modern Human Hunter-Gatherers
This has contributed to widespread, although never universal, agreement that Neanderthals, other than the founding population who evolved into modern humans in Africa, have made almost no contribution to modern human DNA. Neanderthals were considerably more different genetically from the modern human hunter-gatherers who came after them than any two human beings alive today.
On this basis, there is an increasing consensus that even though Neanderthals and modern human hunter-gatherers co-existed in certain areas for thousands of years and almost certainly encountered each other (thousands of years in any one place in the overall time period of circa 50,000 to 25,000 years ago), that they were separate species of hominins.
Perhaps there were some mixed Neanderthal-modern human children, but (1) the number of modern humans was probably greater than the number of Neanderthals (who may have numbered in the single digit thousands in global population), (2) they would have been a small proportion of all children born at the time (due to endogamy involving culturally and ethnically dissimilar populations), and (3) they would probably have been less likely than average to have descendants who survived them (modern humans survived while Neanderthals went extinct for a reason). Further, any slight trace of Neanderthal DNA would also be hard to distinguish based on DNA testing from their genetic link to modern humans, in general, as our closest genetic ancestor.
We have hard DNA evidence that two thousand years ago there were isolated individuals (Westerner in Mongolia, see also Easterner in Italy) with genes that indicate that they have ethic roots from areas thousands of miles away who left no detectable mark on the populations around them. Below some threshold population, the possibility that some minority population can be entirely wiped out and invisible in contemporary populations is not just real but significant.
Modern Human Hunter-Gatherers v. Neolithic Farmers
There also appears to be a growing consensus that modern human hunter-gatherers have made a relatively modest genetic contribution to current modern human populations in Europe. See, e.g., here (paternal descent through Y chromosomes) and here (maternal descent through mitochondrial DNA and shows 18% descent from indigenous hunter-gatherers).
In this interpretation, the spread of farming happened more via colonization by peoples who already farmed than it did by cultural transmission to existing hunter-gatherer populations, perhaps missionary style, perhaps with a thin group of farmers who taught them. Some think that indigenous male lines were less likely to make it into the farmer gene pool than indigenous female lines, a pattern seen in modern times where hunter-gatherer populations intersect with nearby farm populations.
Estimates of the genetic differences between European hunter-gatherers and the Neolithic farmers who overwhelmed or replaced them to a great extent, are on the order of the greatest differences between any two human beings alive today.
It is worth noting that it doesn't take a lot of endogamy (having children within the group) to produce this result. Neolithic farmers almost certainly had a much higher population density than modern human-hunter gatherers.
So, in a colonization scenario, even if these two populations made no ethnic distinctions in child bearing and marriage just a few generations after first contact, the Neolithic farmers would make a far larger contribution than modern human hunter-gatherers who preceded them. Of course, in a colonization scenario, one would expect at least some endogamy within the founding population and within the indigenous population. Indeed, there is every reason from history to believe that war and disease may have decimated indigenous populations that made contact with Neolithic farmer populations, or at least made a dent in them.
I've seen arguments made by published scholars in the field that Neolithic farmers may have been overwhelmingly endogamous for as long as a thousand years after first contact with hunter-gatherers, perhaps with a separation maintained through a caste system, or perhaps because the different societies chose to avoid each other.
The Impact of Settlement Waves On Traces Of Prior Waves
The limited impact of modern human hunter-gatherer populations relative to Neolithic farmer influence on contemporary human genetic makeup also diminishes the likelihood that mixed Neanderthal-modern human children would have contributed significantly to contemporary human genetic makeup.
Neanderthals would have co-existed with modern human hunter-gatherer populations. Neanderthals appear to have gone extinct fifteen thousand years or so, before the Neolithic revolution in which plants and animals were domesticated. Any contribution Neanderthal DNA had to the population genetics of modern human hunter-gatherers (and this has not been established at all), would be diluted in the contemporary population along with the genetic contributions of modern human hunter-gathers generally. This is particularly true if a modern human hunter-gatherer who had Neanderthal DNA was viewed as a poor mating prospect by Neolithic settlers, simply because those individuals would have probably appeared least like members of the Neolithic farmer population.
The dilution of traces of Neanderthal DNA and modern human hunter-gatherer DNA by later waves of settlement would have been even greater if there were multiple waves of settlement by farming populations.
For example, one can imagine a scenario in which modern human hunter-gatherers replace Neanderthals, then Neolithic farmers mostly replace modern human hunter-gatherers, then Indo-European farmers partially replace the first wave of Neolithic farmers, then Roman farmers partially replace Indo-European/Neolithic mix farmers, then invading nomadic pastoralists (i.e. the people the Romans saw as barbarians like the Goths and Visigoths and Huns) partially replaced those populations.
Population replacement in each of those four waves of Neolithic era colonization wouldn't need to be as complete to have a similar effect. And, sexual selection biases against traits found in older populations wouldn't have to appear in every wave, or could be relatively mild in each wave of colonization, yet still have a major impact on the contemporary modern human population.
Scientists looking at existing genetic evidence don't argue that there was near complete replacement of any population after the Neanderthals. But, there is evidence from population genetics, linguistics, remains of material culture, and recorded history that many parts of the world have had multiple waves of settlement.
At least one leading group of population geneticists estimates that there were several major waves of modern huamn population expansion into Europe, the first being a hunter-gatherer wave, and the next being farmers or nomadic pastoralists (i.e. herders), assigning one wave to each of the statistical "principal contributions" to the contemporary genetics of Europe's population and giving it a historical context (see generally regarding principal component analysis of population genetics in Europe).
Linguistic Evidence of Waves of Settlement
There is evidence that people like the Basque of modern Spain, the Estrucians of modern Italy, and the Dravidians of India were farmers before Indo-European language speakers arrived, which would imply that there might have been at least two waves of pre-historic, post-hunter-gatherer colonization. The contemporary populations most closely associated with Europe's hunter-gatherer populations are the nomadic people of Finland and its vincity in the former Soviet Union.
In Europe, one Indo-European wave could have been the Roman Empire itself. Many pre-Latin languages of the Mediterranean and Southern Europe probably died in the face of Roman conquest and centuries of Roman rule. Most of the Indo-European languages that survive in Western Europe, Southern Europe and the Balkans there are descendants of Latin and/or later Germanic and Slavic invaders. But, if there was more than one pre-Roman wave of settlement, pre-Roman peoples could have also spoken Indo-European languages. And, it is possible that Basque and Finnish, for example, are modern human hunter-gatherer languages that survived more than one wave of Indo-European expansion. Genetic evidence suggests the Indo-European languages spread no earlier than the Neolithic revolution.
In India, the Indo-European wave is associated with Vedic culture and languages descended from Sanskrit, while the Dravidian languages are associated with the earlier Indus River Valley civilization (although the ancient writings of this civilization have not been deciphered so its language is not known for sure), and possibly further back still to the Southwestern Iranian language found near Sumer of Elam.
It isn't entirely clear if there were pre-Indo-European languages spoken in the part of Europe never subject to the Roman Empire for long periods of time by Neolithic farmers, although it is likely that European modern human hunter-gatherers did not speak an Indo-European language. We have fairly specific dates for the first appearance of farming in most places in the world, but don't always know what language was originally spoken by those people.
Note also that a couple of prevailing theories about the origins of the Indo-European languages, one placing that origin in Anatolia (i.e. modern Turkey) and another placing that origin with the Kurgan people.
The notion that every Eurasian language must have descended from some language of the Near Eastern people who came out of Africa has a powerful intuitive appeal. Humanity was penned up there until about 50,000 to 60,000 years ago.
So does the notion that language spread together with any given cluster of domesticated plants and animals developed in the Neolithic revolution about 11,000 years ago. Thus, we would expect that Near Eastern farmers and herders would have brought their language with them, while Chinese farmers and herders would have brought a different language with them, on the theory that each farming society would have culturally overwhelmed its hunter-gatherer neighbors.
This still leaves plenty of time for the formation of large numbers of languages. Languages, in isolation, appear to naturally evolve into new languages every 500-1000 years or so.
In a model with multiple waves of settlement, one would expect to see more complete replacement of the earlier populations with less extreme assumptions about relatively population sizes and endogamy.
Multiple waves of settlement also act like an eraser, wiping out evidence of humanity's pre-history more completely than a simple hunter-gatherer replacement by farmers model.
We've seen this phenomena in action in the last two millenia. The late Roman Empire banned "pagan" and non-orthodox monotheistic religious practice and decreed that its artifacts and texts be destroyed. The Islamic empire's expansion in the South and East, and the Roman Catholic Church in the West and North provided a second wave of this purge. As a result, much of the classical pagan and non-orthodox religious lore that existed in 325 CE, when Roman Emperor Constantine declared that Christianity was to be officially tolerated, was obliterated twelve hundred years later and is forever lost.
Thus, while we know that people have had religious practices for tens of thousands of years, we have only a dim view of what this looked like prior to 325 CE, and an even dimmer view of what prehistoric religious beliefs may have involved.
We've also seen many languages die since one of the first historically documented language deaths, that of the Estrucian language, during Roman rule. Many of the dead languages died, or evolved beyond recognition, before they were ever reduced to writing.
The result of this is that the vast majority of contemporary people may have cultural and even genetic roots linking them that are far younger than the emergence of modern humans from Africa. Put another way, most of pre-history was lived by people who left almost no mark on the present. Their languages died. Their religions and histories and culture were lost. Their artifacts were poorly preserved. Their genetic traces in contemporary populations are slight. Pre-history, as well as history itself, is written by the victors.