Ocean tomography [W. Munk and C. Wunsch, Deep Sea Res. 26, 123–61 (1979)] introduces several complications beyond those of conventional inverse theory. The ocean environment represents a significant challenge to the problems of time keeping and data handling. Further, the ocean is a randomly fluctuating medium that changes rapidly in time. These random effects must be either utilized or filtered out by estimation procedures before inverse problems in underwater acoustics can be solved. The random effects are most significant when amplitude ocean acoustic tomography is attempted because micromultipath interference can cause large amplitude discrepancies. A previous attempt to invert for sound-speed structure in a randomly fluctuating ocean failed [M. Brown, U. Calif. San Diego, Ph.D. thesis (1992)]. This paper outlines a successful method for inversion in a randomly fluctuating medium. Simple random inverse problems, such as two-dimensional travel time and one-dimensional waves, are studied. Criteria for stability, uniqueness and model resolution are established. The size of the fluctuations and the extent of the randomness are varied, and an iterative scheme for higher order terms is discussed. The paper provides an analysis of the stochastic forward or direct problem. Parametric and nonparametric estimation procedures are used to solve the inverse problem, and estimator and filter performance are analyzed. Computer solutions to simulated random inverse problems are given. This work is ongoing and will culminate in a successful solution to the random ocean inverse problem.