The drive towards the development of molecular electronics is placing increasing demands on the level of control that must be exerted on the electronic structure of materials. Proposed device architectures ultimately rely on tuning the interactions between individual electronic states, which amounts to controlling the detailed spatial structure of the electronic wavefunctions in the constituent molecules1,2. Few experimental tools are available to probe this spatial structure directly, and the shapes of molecular wavefunctions are usually only known from theoretical investigations. Here we present scanning tunnelling spectroscopy measurements of the two-dimensional structure of individual wavefunctions in metallic single-walled carbon nanotubes; these measurements reveal spatial patterns that can be directly understood from the electronic structure of a single graphite sheet, and which represent an elegant illustration of Bloch's theorem at the level of individual wavefunctions. We also observe energy-dependent interference patterns in the wavefunctions and exploit these to directly measure the linear electronic dispersion relation of the metallic single-walled carbon nanotube.