My research involves the study of operator algebras, operator spaces and quantum information theory. In particular, I am interested in: Reflexivity and hyperreflexivity: a (non-selfadjoint) operator algebra is reflexive if, loosely speaking, it has a lot of invariant subspaces. It is hyperreflexive if the distance to the algebra can be estimated using its invariant subspaces; this is stronger than reflexivity. These properties have natural generalisations from operator algebras to operator spaces, and a key question is to try to determine which operator spaces are reflexive and which are hyperreflexive. Completely bounded mappings between operator spaces and their norms: completely bounded mappings are at the heart of the theory of operator spaces. A linear map between operator spaces comes equipped with two natural norms: the operator norm, and the completely bounded norm. One fundamental problem is to determine when one of these two norms may be estimated using the other, or when they are equal. Schur multipliers: these completely bounded mappings have many attractive properties; for example, their norm and completely bounded norm always agree. However, there are many interesting open questions about this class of mappings, such as: what are the possible values of the norm of an idempotent Schur multiplier? Quantum information theory tackles problems of fundamental importance to the success of quantum computing, a technology still very much in its infancy. There turn out to be intimate connections with the theory of completely bounded maps. My work to date has focussed on privacy and correctability in the infinite dimensional setting.