Classical Thomson scattering - the scattering of low-intensity light by electrons - is a linear process, in that it does not change the frequency of the radiation; moreover, the magnetic-field component of light is not involved. But if the light intensity is extremely high (~1018Wcm-2), the electrons oscillate during the scattering process with velocities approaching the speed of light. In this relativistic regime, the effect of the magnetic and electric fields on the electron motion should become comparable, and the effective electron mass will increase. Consequently, electrons in such high fields are predicted to quiver nonlinearly, moving in figure-of-eight patterns rather than in straight lines. Scattered photons should therefore be radiated at harmonics of the frequency of the incident light, with each harmonic having its own unique angular distribution. Ultrahigh-peak-power lasers offer a means of creating the huge photon densities required to study relativistic, or `nonlinear' (ref. 6), Thomson scattering. Here we use such an approach to obtain direct experimental confirmation of the theoretical predictions of relativistic Thomson scattering. In the future, it may be possible to achieve coherent, generation of the harmonics, a process that could be potentially utilized for `table-top' X-ray sources.