A method is presented for the inversion of waveform data for the three-dimensional distribution of seismic wave velocities. The method is applied to data from the global digital networks (International Deployment of Accelerometers, Global Digital Seismograph Network); the selected data set consists of some 2000 seismograms corresponding to 53 events and 870 paths. The moment tensors of the earthquakes are determined through an iterative procedure which minimizes the corrupting influence of lateral heterogeneity. A global model is constructed for shear wave velocity, expanded up to degree and order 8 in spherical harmonics, and described by a cubic polynomial in depth for the upper 670 km of the earth's mantle. Although no a priori information is incorporated, the model predictions reproduce much of what is known about the dispersion of mantle waves, for example, high phase velocities for shields, low velocities at ridges, and a strong degree 2 pattern for Rayleigh waves. Since the method makes use of complete waveforms, overtone data are also included. It is shown that the model is reproducible in that substantially the same model can be constructed from each half of the total data set considered independently. The model shows that shields and ridges are major features in the depth interval 25--250 km. The ridges of the southern Pacific and the larger shields persist to 350 km, but the SouthEast Indian Rise is underlain by a high-velocity anomaly at this depth, as is much of the Mid-Atlantic Ridge. At 450--650 km the major features are a broad region of high velocities incorporating South America, much of the South Atlantic and parts of West Africa, a broad region of low velocities in the cental and eastern Pacific, high velocities in the western Pacific, and a low-velocity anomaly beneath the Red Sea and the Gulf of Aden. In the absence of a crustal correction, degrees 2 and 3 show a high positive correlation with the geoid; paradoxically, this is largely destroyed when the distribution in crustal thickness is taken into account. Spherical harmonic degrees 4--7 show a significant negative correlation.