We review the physics of pair density wave (PDW) superconductors. We begin with a macroscopic description that emphasizes order induced by PDW states, such as charge density wave, and discuss related vestigial states that emerge as a consequence of partial meting of the PDW order. We review and critically discuss the mounting experimental evidence for such PDW order in the cuprate superconductors, the status of the theoretical microscopic description of such order, and the current debate on whether the PDW is a "mother order" or another competing order in the cuprates. In addition, we give an overview of the weak coupling version of PDW order, Fulde-Ferrell-Larkin-Ovchinnikov states, in the context of cold atom systems, unconventional superconductors, and non-centrosymmetric and Weyl materials.The GLW energy density consistent with time-reversal, parity, space group, and gauge symmetries is (5, 15)The parameters βi depend upon the specific microscopic model. Depending on which values are found for these, one of five possible ground states can be realized. These five phases include the following: the FF-type phase with only one momentum component, the FF * phase which is a bidirectional version of the FF-type phase, the LO type which include pairing with opposite momentum components: these include the unidirectional phase, and the bidirectional-I (II) phases which have a phase factor of 0 (π/2) between the two unidirectional components. These five states give rise to different patterns of induced order, providing a means to distinguish them. We now turn to these induced orders.Induced Order Parameters www.annualreviews.org • The Physics of Pair Density Waves 3 6 Agterberg et al.