Birds, fish, sea turtles, and various other animals have been reported to sense the geomagnetic field and to use it for orientation, navigation, and homing. In recent years, exciting progress has been made towards elucidating the physical and structural basis of this remarkable phenomenon. This paper focuses on the two hypotheses that drive current research into magnetoreception. One proposal relies on the presence of molecules that undergo magnetically anisotropic chemical reactions due to transient formation of a radical pair. The proposed mechanism-essentially a chemical compass-is theoretically well-established and specifically designed behavioral experiments may indeed be interpreted that way, which has sparked a hunt for the molecules and structures in question. The ferrimagnetic transduction hypothesis, on the other hand, draws its plausibility from both theoretical considerations and the fact that magnetite has been detected in sensory neurons, with stable single-domain particles in fish and micrometer-scale clusters of superparamagnetic nanocrystals in birds. We discuss the limitations of our current knowledge and suggest future studies.