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Abstract We present a fast algorithm for solving the eikonal equation in three dimensions, based on the fast marching method. The algorithm is of the order O(N log N), where N is the total number of grid points in the computational domain. The algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain. The method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity. In addition, the method resolves any overturning propagation wavefronts. We begin with the mathematical foundation for solving the eikonal equation using the fast marching method and follow with the numerical details. We then show examples of traveltime propagation through the SEG/EAGE salt model using point-source and planewave initial conditions and analyze the error in constant velocity media. The algorithm allows for any shape of the initial wavefront. While a point source is the most commonly used initial condition, initial plane waves can be used for controlled illumination or for downward continuation of the traveltime field from one depth to another or from a topographic depth surface to another. The algorithm presented here is designed for computing first-arrival traveltimes. Nonetheless, since it exploits the fast marching method for solving the eikonal equation, we believe it is the fastest of all possible consistent schemes to compute first arrivals.