Introduction. One of the main uses of model theory outside of mathematical logic itself has been the introduction in the early sixties of nonstandard analysis by Abraham Robinson (see [18]). He showed how to apply nonstandard models of the appropriate language to a wide variety of problems in analysis. His construction captured the attention of mathematicians because it made the old idea of infinitesimal quantities available to modern mathematics (for a detailed history of the development of these ideas see the last chapter in [18]).

Robinson's original presentation, which relied heavily on the theory of types, has been “cleaned up”, so that today one does not have to be a logician in order to understand and use nonstandard analysis. Nevertheless there are close ties between model theory and developments that have originated from nonstandard practice. The purpose of this paper is to develop one of these ties: we give a general model theoretic theory of approximation which encompasses approximation methods found in both standard and nonstandard settings.