Simple molecules at ultracold temperatures, combined with high-resolution optical spectroscopy, open the door to molecular and fundamental science that is hard to access with other physical systems.  In this talk, we address unique experiments with diatomic molecules trapped in optical lattices, such as observations of deep subradiance and its use for precision measurements, effects of weak magnetic fields on the asymptotic atom-molecule regime, and quantum-state controlled photofragmentation.  We also discuss the prospects for applying this “molecular clock” to discovering new physics.