In mathematical physics, one of the goals in the study of classical  eld theory (essentially, a variational partial di erential equation (PDE)) with gauge symmetries (a.k.a gauge theories) is a precise and rigorous construction of the corresponding reduced phase space [6, 12, 22, 23]: the space of solutions, endowed with the canonical Poisson structure, and quotiented by the gauge transformations. A large number of technical problems stands in the way, including but not limited to describing the solutions of a non-linear PDE as an in nite dimensional space with some kind of smooth structure, specifying a su ciently regular class of functions on this space on which the Poisson structure is well-de ned, explicitly describing the structure of the quotient, or alternatively the structure of the functions invariant under gauge symmetries. The functions in the latter class are referred to as gauge-invariant observables (or invariants). The scope of this Habilitation Thesis is to address some purely geometric problems that arise in the study of gauge-invariant observables and can be attacked using tools from the theory of di erential invariants [30] and the theory of formal integrability of overdetermined PDEs [31, 35]. The main focus is on General Relativity (GR) as a non-trivial representative example, but the perspective is such that the tools used would also apply to other gauge theories, of which Electrodynamics, Yang-Mills, Chern-Simons, Supergravity and many other models used in fundamental theoretical physics and geometry [12], are prominent examples. The papers collected in this Thesis consist of [25], [27], [26], [10], [28], [29], all of which have been published with the exception of [29], which has been submitted for publication to Communications in Mathematical Physics. In the remainder of the Introduction, we give a brief summary of relevant geometric notions(Section 0.1) and summarize the main problems addressed and results obtained in the above papers, grouped by theme in Sections 0.2, 0.3 and 0.4.