First-order perturbation theory is used for fast 2-D computation of traveltimes. For highest efficiency we implement the perturbation integrals into Vidale's finite-difference (FD) eikonal solver. Traveltimes in the unperturbed reference medium are computed with Vidale's method, while perturbed traveltimes in a slightly deviating perturbed medium are obtained by adding a correction Δt to the traveltimes of the reference medium. To compute Δt, raypaths between source and receivers in the reference medium must be known. In Vidale's method traveltimes are computed on a discrete grid assuming local plane wavefronts inside the grid cells. Rays are not determined in this method. Therefore, we suggest approximating rays by ray segments corresponding to the plane wavefronts in each cell. We compute Δt along these segments and obtain initial values for Δt at cell boundaries by linear interpolation between the corner points of the cells. The FD perturbation method can be used for simultaneous computations of traveltime to a number of slightly different models and is, therefore, applicable to prestack velocity estimation techniques. Furthermore, using isotropic reference media the FD perturbation method allows very fast traveltime computations for weakly general anisotropic media. For the computation of traveltimes to a large number of subsurface grid points, the FD perturbation method is about three orders of magnitude faster than classical anisotropic ray-shooting algorithms. Furthermore, we modify Vidale's FD-eikonal solver for elliptically anisotropic media. Using reference media with elliptical anisotropy allows a higher accuracy of the FD perturbation method and let us consider perturbed models of stronger anisotropy. The extension of the FD-perturbation method to 3-D is straightforward. First-order perturbation theory is used for fast 2-D computation of traveltimes. For highest efficiency we implement the pertubaton integrals into Vidale's finite-difference (FD) eikonal solver. Traveltimes in the unperturbed reference medium are computed with Vidale's method, while perturbed traveltimes in a slightly deviating perturbed medium are obtained by adding a correction Δt to the traveltimes of the reference medium. To compute Δt, raypaths between source and receivers in the reference medium must be known. In Vidale's method traveltimes are computed on a discrete grid assuming local plane wavefronts inside the grid cells. Rays are not determined in this method. Therefore, we suggest approximating rays by ray segments corresponding to the plane wavefronts in each cell. We compute Δt along these segments and obtain initial values for Δt at cell boundaries by linear interpolation between the corner points of the cells. The FD perturbation method can be used for simultaneous computations of traveltime to a number of slightly different models and is, therefore, applicable to prestack velocity estimation techniques. Furthermore, using isotropic reference media the FD perturbation method allows very fast traveltime computations for weakly general anisotropic media. For the computation of traveltimes to a large number of subsurface grid points, the FD perturbation method is about three orders of magnitude faster than classical anistropic ray-shooting algorithms. Furthermore, we modify Vidale's FD-eikonal solver for elliptical anistropy allows a higher accuracy of the FD perturbation method and let us consider perturbed models of stronger anistropy. The extension of the FD-perturbation method to 3-D is straightforward.