When electrons pass through a cylindrical electrical conductor aligned in a magnetic field, their wave-like nature manifests itself as a periodic oscillation in the electrical resistance as a function of the enclosed magnetic flux(1). This phenomenon reflects the dependence of the phase of the electron wave on the magnetic field, known as the Aharonov-Bohm effect(2), which causes a phase difference, and hence interference, between partial waves encircling the conductor in opposite directions. Such oscillations have been observed in micrometre-sized thin-walled metallic cylinders(3-5) and lithographically fabricated rings(6-8). Carbon nanotubes(9,10) are composed of individual graphene sheets rolled into seamless hollow cylinders with diameters ranging from 1 nm to about 20 nm. They are able to act as conducting molecular wires(11-18), making them ideally suited for the investigation of quantum interference at the single-molecule level caused by the Aharonov-Bohm effect. Here we report magnetoresistance measurements on individual multi-walled nanotubes, which display pronounced resistance oscillations as a function of magnetic flux We find that the oscillations are in good agreement with theoretical predictions for the Aharonov-Bohm effect in a hollow conductor with a diameter equal to that of the outermost shell of the nanotubes. In some nanotubes we also observe shorter-period oscillations, which might result from anisotropic electron currents caused by defects in the nanotube lattice.