  In this paper, we characterize quasicrystalline interacting topological
phases of matter i.e., phases protected by some quasicrystalline structure. We
show that the elasticity theory of quasicrystals, which accounts for both
"phonon" and "phason" modes, admits non-trivial quantized topological terms
with far richer structure than their crystalline counterparts. We show that
these terms correspond to distinct phases of matter and also uncover
intrinsically quasicrystalline phases, which have no crystalline analogues. For
quasicrystals with internal $\mathrm{U}(1)$ symmetry, we 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. We then extend these ideas much further and address the complete
classification of quasicrystalline topological phases, including systems with
point-group symmetry as well as non-invertible phases. We hence obtain the
"Quasicrystalline Equivalence Principle," which generalizes the classification
of crystalline topological phases to the quasicrystalline setting.
