Nearly two decades ago, the field of atomic physics was revolutionized by the production of degenerate Bose and Fermi gases. These experimental systems, combined with a microscopic description of atomic interactions, have enriched our understanding of emergent collective phenomena, such as superfluidity and the BCS pairing of electrons. The recent production of ultracold gases of polar molecules has similarly opened up entirely new lines of research in quantum chemistry and the quantum simulation of many-body systems. I will describe our recent experiments that produce and study a system of ultracold ground-state molecules in an optical lattice. In particular, we have recently seen the first evidence for long-ranged, dipolar spin-exchange interactions between molecules in different rotational "pseudospin" states. By tuning the strength of dipole-dipole couplings, we have been able to confirm the microscopic description of dipolar interactions in our system. Furthermore, through comparisons to numerical simulations, we have found strong evidence that our system acts as an ideal "quantum simulator" of a specific spin-1/2 Heisenberg XY model. Lastly, I'll discuss future prospects for studies of ultracold polar molecules, including the coherent interplay between long-ranged interactions and the motion of molecules, relevant to the study of supersolidity and itinerant magnetism.