The contemporary state of neutron diffraction of magnetic structures is analyzed from the standpoint of the theory of symmetry of crystals. It is shown that the varied and numerous structures determined in neutron-diffraction studies can be classified and described by the theory of representations of space groups of crystals. This approach is based on expanding the spin density of the crystal in terms of basis functions of the irreducible representations of its space group. Thus the magnetic structure can be specified by the mixing coefficients of the basis functions. Analysis of a large number of different kinds of magnetic structures shows that they arise in the overwhelming majority of cases, in accord with Landau's hypothesis, from a phase transition that follows a single irreducible representation. This means that the number of parameters that fully fix the magnetic structure of an arbitrarily complex crystal is small and equal to the dimensionality of the responsible irreducible representation. This offers great advantages in employing the symmetry approach in deciphering neutron-diffraction patterns of a crystal under study. This is because it reduces the problem of determining a large number of magnetic-moment vectors of the crystal to finding a small number of mixing coefficients. This review presents the fundamentals of such a symmetry analysis of magnetic structures and methods of determining them from neutron-diffraction data. The described method, which is closely allied to Landau's general theory of phase transitions, is illustrated by the most recent neutron-diffraction studies of magnetic structures. They included the so-called multi-$\mathbf{k}$-structures, which are characterized simultaneously by several wave vectors, and structures described simultaneously by several irreducible representations of the space group of the crystal. The article gives a physical explanation of the existence of such structures. The experimental studies of crystal-lattice distortions accompanying the onset of magnetic ordering are reviewed. It is shown how symmetry arguments allow one to determine these distortions as well as the unknown magnetic structure. This review presents in condensed but accessible form the symmetry approach to describing the magnetic structures of crystals and analyzes on this basis the feasibility and degree of reliability of deciphering them by employing the scattering of unpolarized and polarized neutrons.