In recent years, symplectic geometry has emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to study the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I'll describe how one can use this approach to construct an invariant of knots called knot contact homology. I'll then try to give some sense of what this invariant measures, including recently-discovered relations to representations of the knot group, and to mirror symmetry and topological string theory.