The Weinstein conjecture from the 1970s asserts that every Reeb vector field on a compact odd-dimensional manifold has a periodic orbit. This was proved in the three-dimensional case by Taubes in 2006. We describe some recent extensions in the three-dimensional case proved by various authors. These include existence of two or infinitely many periodic orbits in most cases, generic density of periodic orbits, and quantitative closing lemmas. We give an introduction to the tools (from Seiberg-Witten theory and holomorphic curve theory) used to prove such results.