Abstract: Over the last century, group theoretic methods have been used to great effect in the study of low-dimensional manifolds, from the proof that non-trivial knots exist (Tietze, 1908) to the diverse applications of representation space techniques of recent times. The power of these methods is not surprising as low-dimensional manifolds are determined, for the most part, by their fundamental groups. In this talk I will survey how algebraic varieties of representations with values in SL_2(C) have been used to study the topology and geometry of 3-dimensional manifolds including a discussion of two recent applications - the proof of the finite surgery theorem and the study of families of non-zero degree maps between 3-manifolds.