Abstract: With the groundbreaking work of Fulde, Ferrell, Larkin and Ovchinnikov (FFLO), it was realized that superconducting/superfluid order can also break translational invariance, leading to a phase in which the Cooper pairs develop a coherent periodic spatially oscillating structure. Such pair density wave (PDW) superconductivity/superfluidity has become relevant in a diverse range of systems, including cuprates, organic superconductors, heavy-fermion superconductors, cold atoms, and high-density quark matter. In this talk I discuss recent theoretical developments on PDW/FFLO phases. This will include both a discussion of the microscopic origin of such phases in materials with and without inversion symmetry and a phenomenological description of these phases highlighting the role of fractional vortices and thermal fluctuations.