Computational structures---from simple objects like bits and qubits,
to complex procedures like encryption and quantum teleportation---can
be defined using algebraic structures in a symmetric monoidal
2-category. I will show how this works, and demonstrate how the
representation theory of these structures allows us to recover the
ordinary computational concepts. The structures are topological in
nature, reflecting a close relationship between topology and
computation, and allowing a completely graphical proof style that
makes computations easy to understand. The formalism also gives
insight into contentious issues in the foundations of quantum
computing. No prior knowledge of computer science or category theory
will be required to understand this talk.