The travel times of the first arriving seismic waves through any velocity structure can be rapidly computed on a two- or three-dimensional numerical grid by finite-difference extrapolation from point to point. Wavefronts, rather than the traditional rays, are tracked. Head waves are properly treated and shadow zones are filled by the appropriate diffractions. Differences of less than 0.03 per cent are found between the results of this technique and raytracing for a complex, two-dimensional model. This scheme is useful for the windowing of finite-difference calculations to increase computing speed, and promises to aid in earthquake location, tomographic inversion, and Kirchhoff migration in structures that have strong lateral velocity gradients.