Flexoelectricity-the coupling between polarization and strain gradients-is a universal effect allowed by symmetry in all materials. Following its discovery several decades ago, studies of flexoelectricity in solids have been scarce due to the seemingly small magnitude of this effect in bulk samples. The development of nanoscale technologies, however, has renewed the interest in flexoelectricity, as the large strain gradients often present at the nanoscale can lead to strong flexoelectric effects. Here we review the fundamentals of the flexoelectric effect in solids, discuss its presence in many nanoscale systems, and look at potential applications of this electromechanical phenomenon. The review also emphasizes the many open questions and unresolved issues in this developing field.