  This paper reviews recent advances in the study of strongly interacting
systems of dipolar molecules. Heteronuclear molecules feature large and tunable
electric dipole moments, which give rise to long-range and anisotropic
dipole-dipole interactions. Ultracold samples of dipolar molecules with
long-range interactions offer a unique platform for quantum simulations and the
study of correlated many-body physics. We provide an introduction to the
physics of dipolar quantum gases, both electric and magnetic, and summarize the
multipronged efforts to bring dipolar molecules into the quantum regime. We
discuss in detail the recent experimental progress in realizing and studying
strongly interacting systems of polar molecules trapped in optical lattices,
with particular emphasis on the study of interacting spin systems and
non-equilibrium quantum magnetism. Finally, we conclude with a brief discussion
of the future prospects for studies of strongly interacting dipolar molecules.
