The review addresses the problem of the biological effects of weak magnetic fields, which has been a subject of research for about half a century. The spin-chemical radical-pair mechanism is presented as the most prevalent theory in the literature for explaining the observed phenomena. However, the effectiveness of this mechanism is limited by the spin relaxation processes. Estimates of the rate of spin relaxation for a radical in a protein molecule are given, using various channels such as the Zeeman, hyperfine, spin-orbit, and exchange interactions. An analytical solution to the Liouville—von Neumann equation is presented for two electrons and a nucleus, taking into account the effects of relaxation and chemical kinetics. This solution allows one to link the magnitude of the observed effects to the rate of relaxation. Various aspects of this solution are discussed, including the influence of radiofrequency and hypomagnetic fields, the role of quantum entanglement, and the challenges and prospects of the theoretical approach.