The shortest-path method for the calculation of seismic rays and traveltimes in complicated media is compared with two methods for traveltime calculation that are based on a finite-difference solution of the eikonal equation, namely the algorithm of Vidale and a modification by Gray of the algorithm of van Trier and Symes. This comparison consists of two parts: first, the performance of the three algorithms in forward modeling is tested, and second, their application in Kirchhoff migration is demonstrated. The comparison of the forward modeling shows a very close resemblance between the shortest-path method and the eikonal equation solver of Vidale, but quite large differences with Gray's version, which was developed exclusively for application to migration and therefore favors speed at the cost of a possible loss of accuracy. Surprisingly, application of the three methods to the migration of the Marmousi data set shows differences only in small details. The flexibility of the shortest path method allows for application in exploding reflector migration; each reflector can be reconstructed from zero-offset traveltime data by only one forward calculation. Investigation of the effect of smoothing of the slowness model shows that a moderate amount of smoothing delays the traveltimes by about 1 to 2 percent but has no visible effect on the migration results.