Non-Lorentzian Quantum Field Theory and Decoupling Limits of String Theory

This thesis will explore novel five-dimensional field theories with non-Lorentzian scaling properties. These represent a new class of field theories with interesting symmetries, including supersymmetry, but not the usual Lorentz symmetry. However, in contrast to five-dimensional Lorentzian field theories these theories are renormalizable and hence we expect interesting conformal fixed points. In addition these theories are proposed to describe aspects of six-dimensional Lorentzian conformal field theories such as that on the M5-brane.