This review pictures the current state of the problem of propagation of random light fields in optically transparent media having Kerr and thermal nonlinearity. Models are presented of random laser beams and pulses; the correlation properties of the fluctuating light fields and the fundamental theoretical methods of solving the stochastic nonlinear wave equations of quasioptics are described. Spatial instability, self-focusing, and defocusing, the transformation of the spatial statistics of incoherent beams, and the wind refraction of random fields are examined. Quadratic and cubic dispersion, lag of the nonlinear response, non-steady-state nature of the nonlinear polarization, and compression of random pulses are analyzed. Attention is paid to problems of the mutual influence of spatial and temporal fluctuations of the light field in a nonlinear medium.