Modern DNA, in particular maternally inherited mitochondrial DNA (mtDNA), is now routinely used to trace ancient human migration routes and to obtain absolute dates for genetic prehistory. The errors on absolute genetic dates are often large (50% or more) and depend partly on the inherent evolutionary signal in the DNA data, and partly on our imperfect knowledge of the DNA mutation rate. Despite their imprecision, the genetic dates do provide an independent, consistent and global chronology linking living humans with their ancestors. Combining this chronology with archaeological and climatological data, most of our own mtDNA studies during the past decade strongly imply a major role for palaeoclimate in determining conditions for prehistoric migrations and demographic expansions. This paper summarizes our interpretation of the genetic findings, covering the initial and modest spread of humans within Africa more than 100 ka, the striking re-expansion within Africa 60-80 ka, leading ultimately to the out-of-Africa migration of a single, small group which settled in Australia, Eurasia and America during windows of opportunity at least partly dictated by fluctuations in sea-levels and climatic conditions.