Symplectic geometry has recently emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to examine 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 a way to use this approach to construct a rather powerful invariant of knots called "knot contact homology", and discuss its properties. If time permits, I'll also outline a surprising connection to string theory and mirror symmetry.