Extraordinary properties of nonlinear optical propagation processes in the metamaterials which enable coherent energy exchange between ordinary and backward waves of different nature are described. These processes include harmonic generation and three- and four-wave frequency mixing that allow optical parametric amplification and frequency conversion of the light waves. Striking contrast with properties of the counterparts in ordinary materials are shown, which results in sharp increase of energy-conversion efficiency. Particularly, exotic features arise in amplification and generation of contra-propagating short pulses. Novel materials, which enable such processes through support of coexisting electromagnetic or vibration (optical phonons) modes, both with positive and negative phase velocities, are proposed. The concepts of unique ultracompact photonic devices, such as data processing chips, tunable nonlinear-optical mirrors, filters, switches and sensors, are discussed.