In their famous 1927 experiment, Davisson and Germer observed the diffraction of electrons by a periodic material structure, so showing that electrons can behave like waves. Shortly afterwards, Kapitza and Dirac predicted that electrons should also be diffracted by a standing light wave. This Kapitza-Dirac effect is analogous to the diffraction of light by a grating, but with the roles of the wave and matter reversed. The electron and the light grating interact extremely weakly, via the 'ponderomotive potential', so attempts to measure the Kapitza-Dirac effect had to wait for the development of the laser. The idea that the underlying interaction with light is resonantly enhanced for electrons in an atom led to the observation that atoms could be diffracted by a standing wave of light. Deflection of electrons by high-intensity laser light, which is also a consequence of the Kapitza-Dirac effect, has also been demonstrated. But the coherent interference that characterizes wave diffraction has not hitherto been observed. Here we report the diffraction of free electrons from a standing light wave-a realization of the Kapitza-Dirac effect as originally proposed.