  The extraordinary properties of nonlinear optical propagation processes in
double-domain positive/negative index metamaterials are reviewed. These
processes include second harmonic generation, three- and four-wave frequency
mixing, and optical parametric amplification. Striking contrasts with the
properties of the counterparts in ordinary materials are shown. We also discuss
the possibilities for compensating strong losses inherent to plasmonic
metamaterials, which present a major obstacle in numerous exciting
applications, and the possibilities for creation of unique ultracompact
photonic devices such as data processing chips and nonlinear-optical sensors.
Finally, we propose similar extraordinary three-wave mixing processes in
crystals based on optical phonons with negative dispersion.
