A large and growing body of research shows that weak magnetic fields can significantly influence various biological systems, including plants, animals, and humans. However, the underlying mechanisms behind these phenomena remain elusive. It is remarkable that the magnetic energies implicated in these effects are much smaller than thermal energies. Here we review these observations, of which there are now hundreds, and we suggest that a viable explanation is provided by the radical pair mechanism, which involves the quantum dynamics of the electron and nuclear spins of naturally occurring transient radical molecules. While the radical pair mechanism has been studied in detail in the context of avian magnetoreception, the studies reviewed here show that magnetosensitivity is widespread throughout biology. We review magnetic field effects on various physiological functions, organizing them based on the type of the applied magnetic fields, namely static, hypomagnetic, and oscillating magnetic fields, as well as isotope effects. We then review the radical pair mechanism as a potential unifying model for the described magnetic field effects, and we discuss plausible candidate molecules that might constitute the radical pairs. We review recent studies proposing that the quantum nature of the radical pairs provides promising explanations for xenon anesthesia, lithium effects on hyperactivity, magnetic field and lithium effects on the circadian clock, and hypomagnetic field effects on neurogenesis and microtubule assembly. We conclude by discussing future lines of investigation in this exciting new area of quantum biology related to weak magnetic field effects.