The review is devoted to consideration of possible observational consequences of modified gravity theories, suggested for explanation of the contemporary accelerated expansion of the universe. The major attention is paid to F(R)-models. It is shown that in systems with rising energy density high frequency and large amplitude oscillations of the curvature scalar, R(t), are induced. These oscillations lead to the production of elementary particles, which may be observed in the spectra of energetic cosmic rays. In the background of such oscillating solutions gravitation repulsion between finite-size objects becomes possible. Since the Lagrangian is a non-linear function of curvature, equations of motion become of higher (4th) order and exhibit very rich pattern of new physical effects. In particular, the evolution of density perturbations is strongly different from that in General Relativity, amplified due to both parametric resonance and anti-friction phenomena.