In this thesis we consider generalisations of symmetry structures, and their applications to quantum field theories and associated vacuum spaces. The construction of supersymmetry algebras is described in abstract notation. We then consider applications of a further generalisation of symmetry structures, namely categorified groups and algebras. In particular, the links between gauge anomalies as 3-cocyle obstructions and Lie 2-algebras are discussed. Finally, we detail algorithms we have implemented to generate statistics describing the vacuum spaces for supersymmetric non-toric Calabi-Yau quiver gauge theories. A selection of results are presented for low order quivers.