An important technique in 3-manifold topology is to study representations of the fundamental group of a 3-manifold into a Lie group. Under appropriate circumstances, a collection of such representations can be used to cut out an algebraic variety that is well defined up to birational equivalence. This is useful not only because it produces algebro-geometric invariants of 3-manifolds, but also because it allows one to reinterpret certain topological questions as (hopefully easier) questions in algebraic geometry. In this talk I'll describe the basic ingredients of celebrated work of Culler and Shalen that uses these techniques together with Bass-Serre theory to find special surfaces in 3-manifolds. I'll also tell you a bit about the significance of these surfaces, and why 3-manifold topologists are willing to go to such lengths to find them.