In this chapter we give a basic overview of known results regarding Craig interpolation for first-order logic as well as for fragments of first-order logic. Our aim is to provide an entry point into the literature on interpolation theorems for first-order logic and fragments of first-order logic, and their applications. In particular, we cover a range of known refinements of the Craig interpolation theorem, we discuss several important applications of interpolation in logic and computer science, we review known results about interpolation for important syntactic fragments of first-order logic, and we discuss the problem of computing interpolants.

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