The mainstream interpretation of the scientific evidence was that Homo sapiens spread into Eurasia where some of them interbred with Neanderthals. From there they spread to all other continents, including back to Africa, most of them still not carrying any Neanderthal DNA, but that in all continents the few with Neanderthal DNA spread it to everybody else, except in Africa. This exception defies all logic.
Some argue that the Sahara at present is a desert and could not have been crossed by humans. However, the Sahara goes through green phases as well, the last of which only ended around 9,000 years ago, giving humans ample time to travel across. Besides, even as a desert the Sahara has been traversed by humans for at least the past 2,500 years.