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This thesis establishes a generalised setting with which to unify the studyof finite local complexity (FLC) patterns. The abstract notion of a "pattern"is introduced, which may be seen as an analogue of the space group ofisometries preserving a tiling but where, instead, one considers partialisometries preserving portions of it. These inverse semigroups of partialtransformations are the suitable analogue of the space group for patterns withFLC but few global symmetries. In a similar vein we introduce the notion of a\emph{collage}, a system of equivalence relations on the ambient space of apattern, which we show is capable of generalising many constructions applicableto the study of FLC tilings and Delone sets, such as the expression of thetiling space as an inverse limit of approximants.  An invariant is constructed for our abstract patterns, the so calledpattern-equivariant (PE) homology. These homology groups are defined usinginfinite singular chains on the ambient space of the pattern, although we showthat one may define cellular versions which are isomorphic under suitableconditions. For FLC tilings these cellular PE chains are analogous to the PEcellular cochains \cite{Sadun1}. The PE homology and cohomology groups areshown to be related through Poincar\'{e} duality.  An efficient and highly geometric method for the computation of the PEhomology groups for hierarchical tilings is presented. The rotationallyinvariant PE homology groups are shown not to be a topological invariant forthe associated tiling space and seem to retain extra information about globalsymmetries of tilings in the tiling space. We show how the PE homology groupsmay be incorporated into a spectral sequence converging to the \v{C}echcohomology of the rigid hull of a tiling. These methods allow for a simplecomputation of the \v{C}ech cohomology of the rigid hull of the Penrosetilings.

Pattern-Equivariant Homology of Finite Local Complexity Patterns