It is difficult to tell definitively what happened to the Neanderthals. Through years of study, researchers still cannot agree on the dates that Neanderthals and Homo sapiens lived or the number of years they coexisted. The only thing they do seem to agree on is that Neanderthals and Homo sapiens did live at the same time and that there are many possible causes that played a part in the Neanderthal extinction.
Prior to the study of genetics, many researchers believed that the large numbers of Homo sapiens travelling into Neanderthal territories exterminated the Neanderthal. New studies in genetic testing and studies in DNA have led to a surprising find; many people today have 1-4% Neanderthal DNA. This means that most likely there was some mating between Homo sapiens and Neanderthal, though we still do not know how frequent this occurred. The population of Neanderthal was not huge, so it is quite possible that through generations of interbreeding, the genetics from the Neanderthal has just gotten swallowed up by Homo sapiens DNA.
Most researchers do not believe that the extinction happened suddenly, but was a long process. It is thought that Neanderthal lived in groups of 10-15, in Europe, Russia, and Asia in caves, with only population of about 10,000 at their largest. Due to the small groups, it is thought that they probably would not have shared ideas between groups in order to build a stronger society. Early modern humans may have been competing for the same food supply and may have caused the population to die out. Temperature changes from little ice ages may have also played a part in the extinction, causing them to venture out of their usual areas to find a place to live and hunt. Humans had to be resourceful and learn to evolve to survive in these extreme temperatures. A large volcano in Italy, Campi Flegrei, is believed to have caused many Neanderthal and early modern human deaths. The volcano was so large it would have hidden the sun for months, large amounts of ash would