We present two new techniques for the inversion of first-arrival times to estimate velocity structure. These travel-time inversion techniques are unique in that they do not require the calculation of ray paths. First-arrival times are calculated using a finite-difference scheme that iteratively solves the eikonal equations for the position of the wavefront. The first inversion technique is a direct extension of linearized waveform inversion schemes. The nonlinear relationship between the observed first-arrival times and the model slowness is linearized using a Taylor series expansion and a solution is found by iteration. For a series of two-dimensional numerical tests, with and without random noise, this travel-time inversion procedure accurately reconstructed the synthetic test models. This iterative inversion procedure converges quite rapidly and remains stable with further iteration. The second inversion technique is an application of simulated annealing to travel-time topography. The annealing algorithm is a randomized search through model space that can be shown to converge to a global minimum in well-posed problems. Our tests of simulated annealing travel-time topography indicate that, in the presence of less than ideal ray coverage, significant artifacts may be introduced into the solution. The linearized inversion scheme outperforms the nonlinear simulated annealing approach and is our choice for travel-time inversion problems. Both techniques are applicable to a variety of seismic problems including earthquake travel-time tomography, reflection, refraction/wide-angle reflection, borehole, and surface-wave phase-velocity tomography.