Abstract: 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 articlewe explain the quantum mechanical basis of the radical pair mechanism and why it is currently the dominant theory of compassmagnetoreception. We also provide a brief account of two important computational simulation techniques that are used to study themechanism in cryptochrome: spin dynamics and molecular dynamics. At the end, we provide an overview of current research on quantummechanical processes in avian cryptochromes and the computational models for describing them. URL