The completion of the human and other genome projects, and the ongoing development of high-throughput experimental methods for measuring genetic variation, have dramatically changed the scale of information available and the nature of the questions which can now be asked in modern biomedical genetics. Although there is a long history of mathematical modelling in genetics, these developments offer exciting new opportunities and challenges for the mathematical sciences. We focus here on the challenges within human population genetics, in which data document molecular genetic variation between different people. The explosion of data on human variation allows us to study aspects of the underlying evolutionary processes and the molecular mechanisms behind them; the patterns of genetic variation in different geographical regions and the ancestral histories of human populations; and the genetic basis of common human diseases. In each case, sophisticated mathematical, statistical, and computational tools are needed to unravel much of the information in the data, with many of the best methods combining complex stochastic modelling and modern computationally-intensive statistical methods. But the rewards are great: key pieces of scientiﬁc knowledge simply would not have been available by other means.