The review presents the most developed by date hypotheses on the mechanisms of the influence of low-frequency magnetic fields (LFMF) on organisms. Biophysical models that explain the resonance-like responses of biological systems to LFMF with a specific frequency and amplitude are described. Two groups can be distinguished among these models: one considers ions-cofactors of proteins as the primary targets for the LFMF influence, and the other regards the magnetic moments of particles in biomolecules. Attention is paid to the dependence of resonance-like LFMF effects on the cell type. A radical-pair mechanism of the magnetic field's influence on biochemical processes is described with the example of cryptochrome. Conditions for this mechanism's applicability to explain the biological effects of LFMF are given. A model of the influence of LFMF on radical pairs in biochemical oscillators, which can explain the frequency-amplitude efficiency windows of LFMF, is proposed.