Flows of quantum and classical information have a hidden geometrical structure, with processes, systems and correlations described by points, lines and surfaces. The resulting algebraic structures are naturally described using symmetric monoidal bicategories, with the requirement for the information flows to be well-behaved giving rise to dualizability conditions on the n-cells. Our approach allows a variety of quantum and classical protocols to be defined by equations between surface diagrams, such that implementations of these protocols are solutions to these equations in appropriate 2-categories. The choice of 2-category then determines the 'model of computation', or 'theory of physics', in which you would like to work. The approach can be applied to classical encryption, classical secret sharing, quantum teleportation, quantum dense coding, complementary observables and quantum erasure. The bicategorical approach unifies all of these structures, showing them all to be examples of a single, simple algebraic object.