In Chapter One we develop maximum principles for harmonic maps between potentially non-compact manifolds. The first approach uses foliations for studying proper harmonic maps and the second approach assumes the stochastic completeness of the domain and combines these with generalised convex functions. This way we gain new understanding of geometrically defined barriers for the existence of harmonic maps. Motivated by these results we also prove similar statements for non-harmonic maps. In Chapter Two we investigate the moduli space of framed L-twisted Bogomolny equations over Σ x [0,1] motivated by the Kapustin-Witten equations. Therein, we consider moduli spaces with and without prescribed singular behaviour. The nonsingular spaces are in a certain way analogues of the cotangent bundle of a complex Lie group G^c and the singular spaces are related to Hecke modifications.