The Neanderthals died out 27,000 years ago. The exact cause is still debated.
They were adapted to cold weather living. The first theory is that when the weather warmed, the large animals they hunted were no longer around, so they starved to death. The second theory is that they were killed off by the Cro-Magnon (early modern humans). The third theory is a combination of the first two. The latest theory is that the Cro-Magnon were more successful at raising their children to adulthood.
In this theory, the Cro-Magnon over populated areas where they both lived, and used up all the food resources, thereby forcing the Neanderthals into starvation, or pushed the Neanderthals into areas where they did starve to death.