This review of some aspects of the magnetoacoustics of ferro- and antiferromagnetic materials has been written in connection with the 25th anniversary of the rise of this field of physics of magnetic phenomena. Primary attention is paid to relatively new problems that have not been reflected in the existing monographs and reviews. The topic is a group of linear magnetoacoustic effects that manifest spontaneous symmetry breaking caused by magnetic ordering in a system of two coupled fields: the magnetization field $\mathbf{M}(\mathbf{r})$ and the deformation field $u_{ij}(\mathbf{r})$. To some extent these effects are analogous to the Higgs effect in the theory of elementary particles (the Higgs mechanism of the origin of the mass of a particle) or the Meissner effect in the theory of superconductivity. A direct analog of the stated effects is the so-called magnetoelastic gap in the magnon spectrum, while an analog of an accompanying effect is the softening of the quasiacoustic modes interacting with it (up to the vanishing of the corresponding dynamic elastic moduli). However, a characteristic feature of such effects in crystalline (anisotropic) magnetic materials is that they are manifested mainly near points of magnetic (spin-reorientation) phase transitions. This review treats the coupled magnetoelastic waves in ferro- and antiferromagnetic materials of different types that show phase transitions with respect to temperature, magnetic field, or pressure.