This review is primarily devoted to a discussion of isotropic magnetic materials with localized magnetic moments between which the exchange interaction is more complex than in the usual Heisenberg model. They include insulators for which the biquadratic or multispin exchange mechanisms are comparable with the bilinear exchange, conductors which do not satisfy the conditions of validity of the RKKY indirect exchange theory, and the nuclear magnetic material solid helium. Additionally, an analysis is made of high-anisotropy magnetic materials behaving similarly to isotropic non-Heisenberg materials. These anisotropic materials have anomalous properties, compared with the Heisenberg case. Experimental data on these materials and their theoretical interpretation are given. The following topics are discussed: 1) the conditions for a strong non-Heisenberg exchange and non-Heisenberg Hamiltonians; 2) quadrupole ordering and order-proper disorder phase transitions; 3) order-order and order-improper disorder phase transitions; 4) metamagnetism of isotropic materials; 5) normal sequences and ``devil's ladders'' of phase transitions between commensurable structures; 6) canted antiferromagnetism of high-symmetry crystals which do not obey the condition for the existence of relativistic Dzyaloshinskii-type interactions.