In this talk, I will discuss interacting quasicrystalline topological phases of matter i.e., phases protected by some quasicrystalline structure. Espousing the perspective of elasticity theory, I will argue that quasicrystals admit non-trivial quantized topological terms with far richer structure than their crystalline counterparts. These terms, which account for both “phonon” and “phason” modes, correspond to distinct phases of matter, some of which are intrinsically quasicrystalline (have no crystalline analogues). For quasicrystals with internal U(1) symmetry, I will discuss a number of interpretations and physical implications of the topological terms, including constraints on the mobility of dislocations in d = 2 quasicrystals and a quasicrystalline generalization of the Lieb-Schultz-Mattis-Oshikawa-Hastings theorem. Finally, I will discuss the “Quasicrystalline Equivalence Principle,” which generalizes the classification of crystalline topological phases to the quasicrystalline setting.