Carbon nanotubes can be thought of as graphitic sheets with a hexagonal lattice that have been wrapped up into a seamless cylinder. Since their discovery in 1991(1), the peculiar electronic properties of these structures have attracted much attention, Their electronic conductivity, for example, has been predicted(2-4) to depend sensitively on tube diameter and wrapping angle (a measure of the helicity of the tube lattice), with only slight differences in these parameters causing a shift from a metallic to a semiconducting state, In other words, similarly shaped molecules consisting of only one element (carbon) may have very different electronic behaviour, Although the electronic properties of multi-walled and single-walled nanotubes(5-12) have been probed experimentally, it has not yet been possible to relate these observations to the corresponding structure. Here we present the results of scanning tunnelling microscopy and spectroscopy on individual single-walled nanotubes from which atomically resolved images allow us to examine electronic properties as a function of tube diameter and wrapping angle, We observe both metallic and semiconducting carbon nanotubes and find that the electronic properties indeed depend sensitively on the wrapping angle, The bandgaps of both tube types are consistent with theoretical predictions, We also observe van Hove singularities at the onset of one-dimensional energy bands, confirming the strongly one-dimensional nature of conduction within nanotubes.