The Weinstein conjecture states that on any closed manifold, a "Reeb" vector field must have at least one closed orbit. Several years ago, Taubes proved the Weinstein conjecture in dimension 3. I will explain joint work with Hutchings showing that any Reeb vector field on a closed three-manifold has at least two closed orbits. Our proof uses a recently developed three-manifold invariant called "embedded contact homology", which I will also introduce.