Abstract: The rich internal structures of diatomic molecules enable a wide range of experiments in regimes not accessible with atoms. Uses of molecules range from measurement of symmetry-violating effects that probe interesting phenomena in nuclear and particle physics, to the study of highly correlated quantum systems, to the control of novel phenomena in chemical reactions. Despite this broad interest, methods for cooling and trapping molecules have been far less advanced than those for atoms. In particular, direct laser cooling of molecules was long considered infeasible: the same complex internal structure that makes molecules useful also makes laser cooling more difficult. Over the past several years, our group and others have found methods to overcome this obstacle. Now, most of the standard tools of atomic laser cooling and trapping have been demonstrated to work, with appropriate modifications and for certain molecules. In this talk I will review progress in laser cooling and trapping of molecules, and give an outlook for future directions enabled by these rapidly-developing methods.