For a finite group G a permutation of G is a complete mapping of G if the mapping is a permutation, and an orthomorphism of G if the mapping is a permutation. Complete mappings of a finite group G correspond to transversals of the Cayley table M of G, and orthomorphisms of G correspond to permutations of the columns of M that yield latin squares orthogonal to M. Complete mappings and orthomorphisms have been used in constructions of mutually orthogonal sets of latin squares and in constructions of latin squares with particular properties. They and related mappings have also been used in many other algebraic and combinatorial constructions. In this paper we will survey the applications of complete mappings, orthomorphisms, near complete mappings, and near orthomorphisms in the construction of orthogonal latin squares, group sequencings, and neofelds.