When - and how - did Neanderthals depart?

It's becoming widely accepted that the last known Neanderthal population, in what is now Spain, survived until about 37,000 years ago. But what happened to them? DNA from the bones of a child who died 30,000 years ago indicates that, at least in this region, some Neanderthal genes were passed on in more modern humans. In other words, the two types must have interbred, something not shown in modern-day human DNA or in evidence from other sites.