Homo sapiens evolved in Africa. They spread across the world from a migration that started about 60,000 years ago. There were almost certainly earlier migrations (as modern H. sapiens have been around for much longer than that) but they were ultimately unsuccessful, which may be linked to the population bottleneck for humans about 70,000 years ago which indicates a reduction in human population levels.
Humans (members of the Genus Homo) also originally evolved in Africa and spread out over the world from there in a number of waves and further evolved. For instance it seems pretty certain that Neanderthals evolved in Eurasie after ancestors of modern humans (probably H. heidelbergensis) migrated there.
Multi-region does not have any good evidence any more. The DNA mapping of the human genome is pretty conclusive. What tends to confuse people is H. sapiens remains dating before about 60k years ago found elsewhere in the world, Out of Africa does not say that humans did not leave Africa prior to 60,000 years ago, in fact some modern humans could have left Africa as far back a 150,000 years ago. What it says is that all living humans are descended from a migration that started 60,000 years ago, that H. Sapiens evolved in Africa and that any earlier migrations left no living descendants.