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Gauge symmetry invariance is an indispensable aspect of the field-theoretic models in classical and quantum physics. Geometrically this symmetry is often modelled with current groups and current algebras, which are used to capture both the idea of gauge invariance and the algebraic structure of gauge currents related to the symmetry. The Hamiltonian anomaly is a well-known problem in the quantisation of massless fermion fields, originally manifesting as additional terms in current algebra commutators. The appearance of these anomalous terms is a signal of two things: that the gauge invariance of quantised Hamiltonian operators is broken, and that consequently it is not possible to coherently define a vacuum state over the physical configuration space of equivalent gauge connections. In this thesis we explore the geometric and topological origins of the Hamiltonian anomaly, emphasising the usefulness of higher geometric structures. Given this context we also discuss higher versions of the gauge-theoretic current groups. These constructions are partially motivated by the $2$-group models of the abstract string group, and we extend some of these ideas to current groups on the three-sphere $S^3$. The study of the Hamiltonian anomaly utilises a wide variety of tools from such fields as differential geometry, group cohomology, and operator K-theory. We gather together many of these approaches and apply them in the standard case involving the time components of the gauge currents. We then proceed to extend the analysis to the general case with all space-time components. We show how the anomaly terms for these generalised current algebra commutators are derived from the same topological foundations; namely, from the Dixmier-Douady class of the anomalous bundle gerbe. As an example we then compute the full set of anomalous commutators for the three-sphere $S^3$ as the physical space.