Summary: We simulate numerically the full dynamics of Faraday waves in three dimensions for two incompressible and immiscible viscous fluids. The Navier-Stokes equations are solved using a finite-difference projection method coupled with a front-tracking method for the interface between the two fluids. The critical accelerations and wavenumbers, as well as the temporal behaviour at onset are compared with the results of the linear Floquet analysis of K. Kumar and L. S. Tuckerman [J. Fluid Mech. 279, 49–68 (1994; Zbl 0823.76026)]. The finite-amplitude results are compared with the experiments of Kityk et al. [Phys. Rev. E 72, p. 036209 (2005)]. In particular, we reproduce the detailed spatio-temporal spectrum of both square and hexagonal patterns within experimental uncertainty. We present the first calculations of a three-dimensional velocity field arising from the Faraday instability for a hexagonal pattern as it varies over its oscillation period.