This work is concerned with two examples of the interactions between differential geometry and analysis, both related to spinors. The first example is the Dirac operator on conformal spin manifolds with boundary. I aim to demonstrate that the analysis of the Dirac operator is a natural generalisation of complex analysis to manifolds of arbitrary dimension, by providing, as far as possible, elementary proofs of the main analytical results about the boundary behaviour of solutions to the Dirac equation. I emphasise throughout the conformal invariance of the theory, and also the usefulness of the Clifford algebra formalism. The main result is that there is a conformally invariant Hilbert space of boundary values of harmonic spinors, and that the pointwise evaluation map defines a conformally invariant metric on the interior. Along the way, many results from complex analysis are generalised to arbitrary (Riemannian or conformal spin) manifolds, such as the Cauchy integral formula, the Plemelj formula, and the L2-boundedness of the Hilbert transform. The second example concerns the geometry of the twistor operator and the analysis of differential operators arising in twistor theory. I study the differential equations on a complex quadric induced by holomorphic vector bundles on its twistor space. In 4 dimensions, there is already a beautiful example of such a relationship, the Ward correspondence between holomorphic vector bundles trivial on twistor lines, and self-dual connections. There are many generalisations of twistor theory to higher dimensions, but it is not clear how best to generalise the Ward correspondence. Consequently, I focus on 6 dimensional geometry, and one possible generalisation proposed by Atiyah and Hitchin, and investigated by Manin and Minh. I study a number of differential equations produced by this 6 dimensional twistor construction, with a view to reconstructing the holomorphic vector bundle on the twistor space from these equations. While this aim has not been realised, some progress has been made.