Neanderthals became extinct around 40,000 years ago. This timing, based on research published in Nature in 2014, is much earlier than previous estimates, and derives from improved radiocarbon-dating methods analyzing 40 sites from Spain to Russia. Evidence for continued Neanderthal presence in the Iberian Peninsula 37,000 years ago was published in 2017.

Various hypotheses on the causes of Neanderthal extinction implicate:

It seems unlikely that any single one of these hypotheses is sufficient on its own; rather, multiple factors probably contributed to the demise of an already low population.

 

Note:   The above text is excerpted from the Wikipedia article Neanderthal extinction, which has been released under the GNU Free Documentation License.