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Using the symmetry theory we analyze of the flexomagnetic effect in all 90magnetic classes and showed that 69 of them are flexomagnetic. Then we explorehow the symmetry breaking, inevitably present in the vicinity of the surface,changes the local symmetry and thus the form of the flexomagnetic tensors. Allpossible surface magnetic classes (in the number of 19) were obtained from the90 bulk magnetic classes for the surface cuts 001, 010 and 100 types. Itappeared that all 90 bulk magnetic classes become flexomagnetic, piezomagneticand piezoelectric in the vicinity of surface. Using the free energy approach,we show that the flexomagnetic effect leads to a new type offlexo-magnetoelectric (FME) coupling in nanosized and bulk materials, in allspatial regions, where the polarization and (anti)magnetization vectors arespatially inhomogeneous due to external or internal forces. The linear FMEcoupling, proportional to the product of the gradients of (anti)magnetizationand polarization, flexoelectric and flexomagnetic tensors, is significant innanosized ferroelectrics-(anti)ferromagnetics, where gradients of thepolarization and magnetization obligatory exist. The spontaneous FME couplinginduced by the spatial confinement give rise to the size-dependent linearmagnetoelectric coupling in nanosized ferroelectrics-(anti)ferromagnetics.  We show that the flexomagnetic effect may lead to improper ferroelectricityin bulk (anti)ferromagnetics via the linear and nonlinear FME coupling.Inhomogeneous spontaneous polarization is induced by the (anti)magnetizationgradient, which exists in all spatial regions, where polarization varies and(anti)magnetization vector changes its direction. The gradient can be inducedby the surface influence as well as by external strain via e.g. the samplebending.

Linear magnetoelectric coupling and ferroelectricity induced by the flexomagnetic effect in ferroics