In this talk we consider representations of square integrable functions on locally compact Abelian groups using so-called generalized translation invariant (GTI) frames. These systems are a generalization of generalized shift invariant (GSI) systems introduced by Hernandez, Labate and Weiss, and independently, Ron and Shen, where one translates the generators along co-compact (but not necessarily discrete) subgroups. One advantage of studying GSI and GTI systems is that they provide a unified theory for many of the familiar representations, e.g., wavelets, shearlets, and Gabor systems. This talk gives an introduction to generalized translation invariant (GTI) systems on LCA groups. We focus on characterizations of those generators of GTI systems that lead to convenient reproducing formulas. This talk is based on joint work with M.S. Jakobsen (TU Denmark).