One of the most fascinating topics in science is the extinction of the neandertals. After a pretty successful run in Europe during the ice age, they disappeared soon after modern humans arrived on the scene. In the greatest who done it in history, scientists have been trying to figure out for more than a hundred years what happened. Did neandertals get bred into humans? Did humans kill off neandertals? Or did they simply out-compete them gradually?
The interbreeding idea is currently in vogue among lots of anthropologists. But I think they're naive. Looking at some facial reconstructions of neandertals, I'm thinking you'd need some serious beer goggles to go down that road. Personally, I'm not seeing the mating argument as very persuasive.
The outright kill hypothesis doesn't seem that persuasive either, but my feeling is a combination of hostility and simple out-competition is probably what happened. In any case, some exciting DNA research is underway which might reveal whether or not we carry any neandertal genetics in our blood.