This thesis consists of an introduction and three research papers in the general area of geometry and physics. In particular we study 7D supersymmetric Yang-Mills theory and related topics in toric and hypertoric geometry. Yang-Mills theory is used to describe particle interactions in physics but it also plays an important role in mathematics. For example, Yang-Mills theory can be used to formulate topological invariants in differential geometry. In Paper I we formulate 7D maximally supersymmetric Yang-Mills theory on curved manifolds that admit positive Killing spinors. For the case of Sasaki-Einstein manifolds we perform a localisation calculation and find the perturbative partition function of the theory. For toric Sasaki-Einstein manifolds we can write the answer in terms of a special function that count integer lattice points inside a cone determined by the toric action. In Papers II and III we consider 7D maximally supersymmetric Yang-Mills theory on hypertoric 3-Sasakian manifolds. We show that the perturbative partition function can again be formulated in terms of a special function counting integer lattice points in a cone, similar to the toric case. We also present a factorisation result for these functions.