The flexoelectric effect is the response of electric polarization to a mechanical strain gradient. It can be viewed as a higher-order effect with respect to piezoelectricity, which is the response of polarization to strain itself. However, at the nanoscale, where large strain gradients are expected, the flexoelectric effect becomes appreciable. Besides, in contrast to the piezoelectric effect, flexoelectricity is allowed by symmetry in any material. Due to these qualities flexoelectricity has attracted growing interest during the past decade. Presently, its role in the physics of dielectrics and semiconductors is widely recognized and the effect is viewed as promising for practical applications. On the other hand, the available theoretical and experimental results are rather contradictory, attesting to a limited understanding in the field. This review paper presents a critical analysis of the current knowledge on the flexoelectricity in common solids, excluding organic materials and liquid crystals.