The advent of the laser as an intense source of coherent light gave rise to nonlinear optics, which now plays an important role in many areas of science and technology. One of the first applications of nonlinear optics was the multi-wave mixing of several optical fields in a nonlinear medium (one in which the refractive index depends on the intensity of the field) to produce coherent light of a new frequency. The recent experimental realization of the matter-wave 'laser' - based on the extraction of coherent atoms from a Bose- Einstein condensate - opens the way for analogous experiments with intense sources of matter waves: nonlinear atom optics. Here we report coherent four- wave mixing in which three sodium matter waves of differing momenta mix to produce, by means of nonlinear atom-atom interactions, a fourth wave with new momentum. We find a clear signature of a four-wave mixing process in the dependence of the generated matter wave on the densities of the input waves. Our results may ultimately facilitate the production and investigation of quantum correlations between matter waves.