Over the last 50 years, the study of the properties of neutrinos has unveiled a number of surprising facts that necessitate physics beyond the standard model. We now know that neutrinos are not only massive, but that there is a non-trivial alignment between the mass and flavour bases, inducing flavour changing transitions known as neutrino oscillations. Understanding the neutrino sector is a crucial first step in our attempts to extend our current theories of fundamental physics, and studies of neutrino oscillation provide us with a unique tool to probe these elusive particles. In this thesis, we assess the potential of the next generation of neutrino oscillation experiments to probe physical effects both within and beyond the current neutrino flavour paradigm: resolving existing unknowns, and constraining the correlations induced by theories of leptonic flavour.