The ocean stores and transports vast quantities of heat, fresh water, carbon and other materials, and its circulation plays an important role in determining both the Earth's climate and fundamental processes in the biosphere. Understanding the development of climate and important biological cycles therefore requires detailed knowledge of ocean circulation and its transport properties. This cannot be achieved solely through modelling, but must involve accurate observations of the spatio-temporal evolution of the global oceanic flow field. Estimates of oceanic flow are currently made on the basis of space-borne measurements of the sea surface, and monitoring of the ocean interior. Satellite altimetry and acoustic tomography are complementary for this purpose1, as the former provides detailed horizontal coverage of the surface, and the latter the requisite vertical sampling of the interior. High-quality acoustic-tomographic2 and altimetric3 data are now available to test the combined power of these technologies for estimating oceanic flows. Here we demonstrate that, with the aid of state-of-the-art numerical models, it is possible to recover from these data a detailed spatio-temporal record of flow over basin-scale volumes of fluid. Our present results are restricted to the Mediterranean Sea, but the method described here provides a powerful tool for studying oceanic circulation worldwide.