The highest court of the United Nations, the International Court of Justice, has recognized the legality of Kosovo's February 17, 2008 Declaration of Independence from Serbia. Since then, sixty-nine countries, including the United States, Canada, Japan, Australia and most of the countries of Western Europe, have recognized Kosovo an a sovereign independent state over Serbia's objections supported mostly by Russia and other formerly communist countries.
Saudi Arabia and many other predominantly Muslim countries have recognized Kosovo's independence, in part because they feel that Kosovo is defending a Muslim population which was minority in Serbia.
The International Court of Justice ruling could help Kosovo win recognition from other countries, which covets a United Nations membership.
The U.N. ruling also gives heart to those separatist regions who would like to unilaterally declare independence. Previously rulings on the circumstances in which a subdivision of an existing state may unilaterally declare independence consistent with international law, such as a Supreme Court of Canada ruling addressing the conditions under which Quebec could become an independent country, have tended to require either consent from the country that the region used to belong to, or widespread international recognition. (As an aside, the United Kingdom long ago formally recognized the legality of the separation of the United States from it in a treaty.)
Kosovo isn't the only country of disputed sovereignty where sovereignty is recognized by some states but not others, in the world. Most are unsettled fallout from the collapse of the Soviet Union in the Caucuses or are Cold War hold overs (North Korea and South Korea, Taiwan and the People's Republic of China). Israel-Palestine, Cyprus-Northern Cyprus, Morocco-Western Sahara, and Somalialand-Somolia disputes also remain outstanding.
Of the currently live disputes, those in the Caucuses, in Somolia and in Kosovo, however, are the only situations likely to be in flux in the near future. The Chinese, Korean, Israeli, Cyprus and Morocco situations all appear to have reached stalemates for the time being.
Notably, Turkey has recognized the independence of Kosovo, despite the possible relevance of that recognition of a unilateral declaration of independence by an ethnic minority region to its own dispute with the Kurds.
Some of the notable European states that have not yet recognized Kosovo are Spain and Greece. In the case of Spain, this may be motivated by concerns about declarations of independence by its only ethnically distinct autonomous regions (Basque and Catalan communities, for example).