Abstract: We discuss motivations for modifying gravity, review the most typical extensions of Einstein-Hilbert gravity, and present the most important results obtained within these extensions, including the cosmological context, and paying the essential attention to the problem of Lorentz symmetry breaking in gravity. Actually, there are three main directions in studying of Lorentz-breaking extensions of gravity: first, restricting to the case of weak (linearized) gravitational field where solving the problem of consistency of the Lorentz symmetry breaking with the gauge symmetry is easier, second, consideration of the Einstein gravity with the additive four-dimensional grav- itational Chern-Simons term which breaks parity, and, for special choice of the Chern-Simons coefficient, breaks also the Lorentz symmetry, third, using the mechanism of spontaneous Lorentz symmetry breaking. Within our talk, we present the main results achieved by us within these approaches.