We report the results of an extensive numerical study of the small-scale turbulent dynamo. The primary focus is on the case of large magnetic Prandtl numbers Pr. A Pr scan is given for the case of viscosity-dominated (low-Reynolds-number) turbulence. We concentrate on three topics: magnetic-energy spectra and saturation levels, the structure of the magnetic-field lines, and intermittency of the field-strength distribution. The main results are (i) the folded structure of the field (direction reversals at the resistive scale, field lines curved at the scale of the flow) persists from the kinematic to the nonlinear regime, (ii) the field distribution is self-similar and appears to be lognormal during the kinematic regime and exponential in the saturated state; (iii) the bulk of the magnetic energy is at the resistive scale in the kinematic regime and remains there after saturation, although the magnetic-energy spectrum becomes much shallower. We propose an analytical model of saturation which predicts saturated spectra in excellent agreement with numerical results. Comparisons with large-Re, moderate-Pr runs are carried out to confirm the relevance of these results and to test heuristic scenarios of saturation. New features at large Re are elongation of the folds in the nonlinear regime from the viscous scale to the box scale and the presence of an intermediate nonlinear stage of slower-than-exponential growth accompanied by an increase of the resistive scale and partial suppression of the kinetic-energy spectrum in the inertial range. Numerical results for the saturated state do not support scale-by-scale equipartition between magnetic and kinetic energies, with a definite excess of magnetic energy at small scales. A physical picture of the saturated state is proposed.