Traditionally, philosophers of quantum mechanics have addressed exceedingly simple systems: a pair of electrons in an entangled state, or an atom and a cat in Dr. Schrodinger's diabolical device. But recently, much more complicated systems, such as quantum fields and the infinite systems at the thermodynamic limit of quantum statistical mechanics, have attracted, and repaid, philosophical attention. Interpreting Quantum Theories has three entangled aims. The first is to guide those familiar with the philosophy of ordinary QM into the philosophy of 'QM infinity', by presenting accessible introductions to relevant technical notions and the foundational questions they frame. The second aim is to develop and defend answers to some of those questions. Does quantum field theory demand or deserve a particle ontology? How (if at all) are different states of broken symmetry different? And what is the proper role of idealizations in working physics? The third aim is to highlight ties between the foundational investigation of QM infinity and philosophy more broadly construed, in particular by using the interpretive problems discussed to motivate new ways to think about the nature of physical possibility and the problem of scientific realism.