Let T(w,x) = y( w ) be a nonlinear random operator equation with not necessarily unique solution. For this and similar equations, we prove results about convergence of solutions of suitable approximate problems T_n(w,x) =Y_n(w) to solutions of the original equations. We do this for rather general notions of convergence for random variables. Concepts like consistency, stability, and compactness in sets of measurable functions are introduced and used. For all assumptions that are needed in the general theory, sufficient conditions are given with respect to convergence in probability and almost-sure convergence. As a specific method for constructing approximate equations we discuss "discretization schemes ", where the underlying probability space is discretized. Some results might be of interest also in different contexts; these include criteria for almost-sure convergence of measurable multifunctions and results about compactness with respect to convergence in probability and almost-sure convergence.