Pattern-equivariant cohomology theory was developed by Ian Putnam and Johannes Kellendonk in 2003, for tilings whose tiles appear in fixed orientations. In this dissertation, we generalize this theory in two ways: first, we define this cohomology to apply to tiling spaces, rather than individual tilings. Second, we allow tilings with tiles appearing in multiple orientations - possibly infinitely many. Along the way, we prove an approximation theorem, which has use beyond pattern-equivariant cohomology. This theorem states that a function which is a topological conjugacy can be approximated arbitrarily closely by a function which preserves the local structure of a tiling space. The approximation theorem is limited to translationally finite tilings, and we conjecture that it is not true in the infinite case.

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