Recent studies of optical, magneto-optical, and nonlinear optical effects in hyperbolic metamaterials based on metal nanorods in a dielectric template are reviewed. An increase in the efficiency of the optical second harmonic generation in the spectral vicinity of zero effective permittivity (epsilon near zero) of hyperbolic metamaterials, as well as a jump of the phase of a quadratic nonlinear optical signal, is detected at the transition between elliptic and hyperbolic dispersion regimes. The resonant enhancement of magneto-optical effects in composite magnetic hyperbolic metamaterials in the spectral vicinity of nearly zero effective permittivity is demonstrated. Effects of fast and slow light in the interaction of ultrashort laser pulses with hyperbolic metamaterials are experimentally revealed.