  The magnetoelectricity of garnets is considered by means of a symmetry and
quantum mechanical combined analysis. It is shown, that the magnetoelectric
effect is not realized in most garnets although the necessary condition of the
crystal magnetic structure antisymmetry in them is held at low temperatures.
Nevertheless, the effect can be observed in some garnets as well as other odd
effects, namely, piezomagnetic effect, magnetic field evoked piezoelectric one,
etc. It is also discovered that magnetic fields can induce specific
antiferroelectric structures in garnet crystals and produce electric
polarization in epitaxial films. The polarization can also be caused in a bulk
crystal by an inhomogeneous magnetic field.
