A review is given of some recently obtained results on analytic contractions of Lie algebras and Lie groups and their application to special function theory. The contractions considered are from O(3) to E(2) and from O(2,1) to E(2), or E(1,1). The analytic contractions provide relations between separable coordinate systems on various homogeneous manifolds. They lead to asymptotic relations between basis functions and overlap functions for the representations of different groups.