The Out of Africa or African Replacement Hypothesis is a well-supported theory that argues that every living human being is descended from a small group in Africa, who then dispersed into the wider world displacing earlier forms such as Neanderthal. Early major proponents of this theory were led by Chris Stringer. The Out-of-Africa theory was bolstered in the early 1990s by research on mitochondrial DNA studies by Allan Wilson and Rebecca Cann which suggested that all humans ultimately descended from one female: the Mitochondrial Eve.
Today, the vast majority of scholars have accepted that human beings evolved in Africa and migrated out; recent evidence has shown that happened in multiple waves. The number and timing of the waves is still being debated.
Scholars largely agree that our modern species (Homo sapiens) originated in east Africa by 195-160,000 years ago. The earliest known pathway Out of Africa probably occurred between Marine Isotope Stage 5e, or between 130,000-115,000 years ago, along the Nile Corridor and into the Levant, evidenced by Middle Paleolithic sites at Qazfeh and Skhul. That migration (sometimes confusingly called "Out of Africa 2" because it was discovered more recently than the next) is generally regarded as a "failed dispersal", because only a handful of Homo sapiens sites have been identified as being this old outside of Africa. However, fossil evidence of any kind this old is pretty rare and it may be too early to completely rule that out.
A later pulse from northern Africa, which was recognized at least thirty years ago, occurred from about 65-40,000 years ago [MIS 4 or early 3], through Arabia: that one, scholars believe, eventually led to the human colonization of Europe and Asia, and the eventual replacement of Neanderthals in Europe.
The fact that these two pulses occurred in the past are largely undebated today. A third, and increasingly convincing, human migration is the southern dispersal hypothesis, which argues that an additional wave of colonization occurred between those two better-known pulses. Growing archaeological and genetic evidence supports the existence of this earlier southern route into South Asia.