Numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media, and they promise to be invaluable in solving the full inverse problem. This paper quantitatively compares finite-difference and finite-element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time-domain and frequency-domain techniques. It is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy. To be of value, a solution technique must be able to minimize (1) numerical attenuation or amplification, (2) polarization errors, (3) numerical anisotropy, (4) errors in phase and group velocities, (5) extraneous numerical (parasitic) modes, (6) numerical diffraction and scattering, and (7) errors in reflection and transmission coefficients. This paper shows that in homogeneous media the explicit finite-element and finite-difference schemes are comparable when solving the scalar wave equation and when solving the elastic wave equations with Poisson's ratio less than 0.3. Finite-elements are superior to finite-differences when modeling elastic media with Poisson's ratio between 0.3 and 0.45. For both the scalar and elastic equations, the more costly implicit time integration schemes such as the Newmark scheme are inferior to the explicit central-differences scheme, since time steps surpassing the Courant condition yield stable but highly inaccurate results. Frequency-domain finite-element solutions employing a weighted average of consistent and lumped masses yield the most accurate resuls, and they promise to be the most cost-effective method for CDP, well log, and interactive modeling.--Modified journal abstract.