A review is made of theoretical concepts from nonlinear dynamics of quasi-one-dimensional magnetic materials and of experimental investigations of solitons by inelastic neutron scattering and of studies of anomalies of thermodynamic quantities such as the specific heat, magnetization, susceptibility, etc. The main investigated substances are the quasi-one-dimensional ferromagnet CsNiF$_3$ and the antiferromagnet tetramethylammonium manganese chloride (TMMC) which share the same crystal structure. They can be regarded as quasi-one-dimensional magnetic materials with easy-plane anisotropy. The spin dynamics of such a system subjected to an external magnetic field applied in the easy plane can be reduced to the sine-Gordon equation. A detailed analysis is given of the recent experiments on CsNiF$_3$ and TMMC carried out using unpolarized and polarized neutrons and demonstrating that in a certain range of temperatures and fields this dynamics includes nonlinear excitations which can be described qualitatively as solitons of the sine-Gordon equation. Another group of quasi-one-dimensional crystals with easy-axis anisotropy is considered: it belongs to Ising-like magnetic materials. In the absence of an external field these materials exhibit soliton-type nonlinear excitations in the form of antiphase domain walls. The available experimental data confirm the concept of a soliton gas of excitations in quasi-one-dimensional magnetic materials.