The growth and saturation of magnetic field in conductingturbulent media with large magnetic Prandtl numbers areinvestigated. This regime is very common in low-density hotastrophysical plasmas. During the early (kinematic) stage, weakmagnetic fluctuations grow exponentially and concentrate at theresistive scale, which lies far below the hydrodynamic viscousscale. The evolution becomes nonlinear when the magnetic energyis comparable to the kinetic energy of the viscous-scale eddies. A physical picture of the ensuing nonlinear evolution of the MHD dynamo is proposed. Phenomenological considerations are supplemented with a simple Fokker-Planck model of thenonlinear evolution of the magnetic-energy spectrum. It isfound that, while the shift of the bulk of the magnetic energy from the subviscous scales to the velocity scales may bepossible, it occurs very slowly - at the resistive, ratherthan dynamical, timescale (for galaxies, this means thatthe generation of large-scale magnetic fields cannot be explainedby this mechanism). The role of Alfvénic motions and theimplications for the fully developed isotropic MHD turbulenceare discussed.