The remarkable ability of migratory birds to navigate accurately using the geomagnetic field for journeys of thousandsof kilometres is currently thought to arise from radical pair reactions inside a protein called cryptochrome. In this article we explain the quantum mechanical basis of the radical pair mechanism and why it is currently the dominant theory of compass magnetoreception. We also provide a brief account of two important computational simulation techniques that are used to study the mechanism in cryptochrome: spin dynamics and molecular dynamics. At the end, we provide an overview of current research on quantum mechanical processes in avian cryptochromes and the computational models for describing them.