Abstract: In dimension 3, there is a famous theorem due to Jaco and Shalen and to Johannson asserting the existence and uniqueness of a very special decomposition for any compact 3-manifold. This is described by cutting along a finite family of disjoint embedded annuli and tori. The object of this talk is to describe joint work with Gadde Swarup of the University of Melbourne in which we show that there is an analogous (but far more general) result for decompositions of arbitrary finitely presented groups. In the special case when one applies our decomposition to the fundamental group of a 3-manifold, our algebraic decomposition is precisely equivalent to the classical topological one.