The propagation of an optical soliton is studied in a fiber with randomly varying core diameter or random amplification along the fiber. The adiabatic dynamics of the soliton and radiative processes are investigated by a perturbation method based on the Inverse Scattering Transform (IST). The mean emitted power and the damping rate of the soliton are calculated. The interaction of solitons in random media is investigated using the Karpman-Solov'ev perturbation theory. Numerical simulations of the full nonlinear Schr{\"o}dinger equation with multiplicative white and colored noise perturbations are performed for initial conditions corresponding to a single soliton and two interacting solitons. The results obtained are in good agreement with the analytical estimates for white noise in the initial stage of the propagation. The numerical simulations reveal a new phenomena where single solitons disintegrate or split under white noise perturbation, but stabilize under colored noise action. Finally the existence of a bound state of two interacting solitons is observed in the numerical simulations in media with colored noise perturbation.