The electron-phonon problem has long been considered basically “solved” by a combination of Migdal-Eliashberg theory and Holstein small polaron theory in the weak and strong coupling limits respectively. However, we have recently realized that there are multiple regimes of parameter space in which intuitions based on these long-standing ideas are inapplicable, and in which a variety of new regimes, and even phases of matter can arise. I will discuss some of the progress along these lines, including a set of controlled solutions to model problems in which an exotic phase with deconfined fractionalized excitations and emergent photons arises in an electronically uninteresting “band insulator.”