The conventional way to estimate a distribution function is to assume it belongs to a class parameterized by a finite-dimensional vector and then estimate the unknown parameter vector. In many cases, e.g., regression models, part of the assumption is of the form: a given function of the data and of the parameter vector has a zero mean. We consider estimating distribution functions using only assumptions of this type (moment restrictions). We do not assume that the distribution function belongs to a finite-dimensional parametric class. The motivation for this exercise is that moment restrictions are often implied by theory (a good example is asset pricing models), but distributional assumptions typically are not.