This thesis investigates Hitchin functionals and $h$-principles for stable forms on oriented manifolds, with a special focus on $G2$ and $G~2$ 3- and 4-forms. Additionally, it introduces two new spectral invariants of torsion-free $G2$-structures.

Part I begins by investigating an open problem posed by Bryant, $viz.$ whether the Hitchin functional $H3$ on closed $G2$ 3-forms is unbounded above. Chapter 3 uses a scaling argument to obtain sufficient conditions for the functional $H3$ to be unbounded above and applies this result to prove the unboundedness above of $H3$ on two explicit examples of closed 7-manifolds with closed $G2$ 3-forms. Chapter 3 then proceeds to interpret this unboundedness geometrically, demonstrating an unexpected link between the functional $H3$ and fibrations, proving that the 'large volume limit' of $H3$ in each case corresponds to the adiabatic limit of a suitable fibration. The proof utilises a new, general collapsing result for singular fibrations between orbifolds, without assumptions on curvature, which is proved in Chapter 4. Chapter 5 broadens the focus of Part I to include the Hitchin functionals $H4$, $H~3$ and $H~4$ on closed $G2$ 4-forms, $G~2$ 3-forms and $G~2$ 4-forms respectively. In its main result, Chapter 5 proves that $H4,H~3,H~4$ are always unbounded above and below (whenever defined), and also that $H3$ is always unbounded below (whenever defined). As scholia, the critical points of the functionals $H4$, $H~3$ and $H~4$ are shown to be saddle points, and initial conditions of the Laplacian coflow which cannot lead to convergent solutions are shown to be dense. Part I ends with a short discussion of open questions, in Chapter 6.

Part II investigates relative $h$-principles for closed, stable forms. After establishing some prerequisite algebraic results, Chapter 7 begins by proving that if a class of closed, stable forms satisfies the relative $h$-principle, then its corresponding Hitchin functional is automatically unbounded above. By utilising the technique of convex integration, Chapter 7 then obtains sufficient conditions for a class of closed, stable forms to satisfy the relative $h$-principle, a result which subsumes all previously established $h$-principles for closed stable forms. Until now, 12 of the 16 possible classes of closed stable forms have remained open questions with regard to the relative $h$-principle. In the main result of Part II, Chapters 7 and 8 prove the relative $h$-principle in 5 of these open cases. The remaining 7 cases are addressed in the final chapter of Part II, where it is conjectured that the relative $h$-principle holds in each case. Chapter 9 applies the $h$-principles established in this thesis to prove various results on the topological properties of closed $G~2$, $\mathrm{SL}(3;C)$ and $\mathrm{SL}(3;R)2$ forms. Firstly, it characterises which oriented 7-manifolds admit closed $G~2$ forms, in the process introducing a new technique for proving the vanishing of natural cohomology classes on non-closed manifolds. Next, it introduces $G~2$-cobordisms of closed $\mathrm{SL}(3;C)$ and $\mathrm{SL}(3;R)2$ 3-forms and proves that homotopic forms are $G~2$-cobordant. Additionally, Chapter 9 classifies $\mathrm{SL}(3;C)$ 3-forms up to homotopy and provides a partial classification result on homotopy classes of $\mathrm{SL}(3;R)2$ 3-forms. Part II ends with a short discussion of open questions, in Chapter 10.

Part III introduces and examines two new spectral invariants of torsion-free $G2$-structures. Although the notion of an invariant is a central theme in geometry and topology, currently, there is only one known invariant of torsion-free $G2$-structures: the $\nu ―$-invariant of Crowley-Goette-Nordström. Part III defines two new invariants of torsion-free $G2$-structures, termed $\mu 3$- and $\mu 4$-invariants, by regularising the classical notion of Morse index for the Hitchin functionals $H3$ and $H4$ at their critical points. In general, there is no known way to compute $\nu ―$ for $G2$-manifolds constructed via Joyce's `generalised Kummer construction'. Chapter 11 obtains closed formulae for $\mu 3$ and $\mu 4$ on the orbifolds used in Joyce's construction, leading to a conjectural discussion in Chapter 12 of how to compute $\mu 3$ and $\mu 4$ on Joyce's manifolds.