This thesis is divided into three sections.

The first begins with a review of the Fefferman-Graham embedding space construction and the closely-related tractor calculus, which provides a simple way to catalogue and invent conformally invariant operators. In particular, the conformal wave operator arises as the descendant of the ordinary wave operator on the Fefferman-Graham space. We construct a worldline action on the Fefferman-Graham space whose equations of motion descend to a conformally coupled scalar on the base manifold. We present also a novel Fefferman-Graham worldline action for the massless Dirac equation and demonstrate that it descents to a Dirac spinor on the base manifold. Unlike the scalar, the massless Dirac equation is conformally invariant without modification.

The second concerns the application of twistor theory to five-dimensional anti-de Sitter space. The twistor space of AdS_5 is the same as the ambitwistor space of the four-dimensional conformal boundary; the geometry of this correspondence is reviewed for both the bulk and boundary. A Penrose transform allows us to describe free bulk fields, with or without mass, in terms of data on twistor space. Explicit representatives for the bulk-to-boundary propagators of scalars and spinors are constructed, along with twistor action functionals for the free theories. Evaluating these twistor actions on bulk-to-boundary propagators is shown to produce the correct two-point functions.

In the final chapter, we construct a minitwistor action for Yang--Mills--Higgs theory in three dimensions. The Feynman diagrams of this action will construct perturbation theory around solutions of the Bogomolny equations in much the same way that MHV diagrams describe perturbation theory around the self--dual Yang Mills equations in four dimensions. We also provide a new formula for all tree amplitudes in YMH theory (and its maximally supersymmetric extension) in terms of degree d maps to minitwistor space. We demonstrate its relationship to the RSVW formula in four dimensions and show that it generates the correct MHV amplitudes at d=1 and factorizes correctly in all channels for all degrees.