Migratory birds travel spectacular distances each year, navigating and orienting by a variety of means, most of which are poorly understood. Among them is a remarkable ability to perceive the intensity and direction of the Earth's magnetic field. Biologically credible mechanisms for the detection of such a weak field (25-65 mT) are scarce and in recent years just two proposals have emerged as frontrunners. One, essentially classical, centers on clusters of magnetic iron-containing particles in the upper beak which appear to act as a magnetometer for determining geographical position. The other relies on the quantum spin dynamics of transient photoinduced radical pairs. Originally suggested by Schulten in 1978 as the basis of the avian magnetic compass sensor, this mechanism gained support from the subsequent observation that the compass is light-dependent. The radical pair hypothesis began to attract increased interest following the proposal in 2000 that free radical chemistry could occur in the bird's retina initiated by photoexcitation of cryptochrome, a specialized photoreceptor protein. In the present paper we review the important physical and chemical constraints on a possible radical-pair-based compass sensor and discuss the suggestion that radical pairs in cryptochromes might provide a biological realization for a magnetic compass.