  We put the theory of interacting topological crystalline phases on a
systematic footing. These are topological phases protected by space-group
symmetries. Our central tool is an elucidation of what it means to "gauge" such
symmetries. We introduce the notion of a "crystalline topological liquid", and
argue that most (and perhaps all) phases of interest are likely to satisfy this
criterion. We prove a Crystalline Equivalence Principle, which states that in
Euclidean space, crystalline topological liquids with symmetry group $G$ are in
one-to-one correspondence with topological phases protected by the same
symmetry $G$, but acting *internally*, where if an element of $G$ is
orientation-reversing, it is realized as an anti-unitary symmetry in the
internal symmetry group. As an example, we explicitly compute, using group
cohomology, a partial classification of bosonic symmetry-protected topological
(SPT) phases protected by crystalline symmetries in (3+1)-D for 227 of the 230
space groups. For the 65 space groups not containing orientation-reversing
elements (Sohncke groups), there are no cobordism invariants which may
contribute phases beyond group cohomology, and so we conjecture our
classification is complete.
