ABSTRACT: Nielsen fixed point theory is the study of the minimization of the fixed point set of a mapping when the mapping is changed by a homotopy. Basic techniques were developed by Nielsen and Reidemeister, and arose in many cases from a search for a converse to the Lefschetz fixed point theorem (is there a numerical invariant which is zero if and only if the mapping is homotopic to a fixed point free map?). We will discuss the foundations of the theory, as well as areas of current interest. We will also mention some closely related Nielsen-type theories outside of fixed point theory.