Transient solutions of the transverse depleted-pump and Stokes field distributions are obtained for an externally-pumped solid-state Raman laser. The results are shown to agree well with experiment measurements made on a Q-switched Nd:YAG pumped solid-state barium nitrate Raman laser. The theory of intracavity Raman conversion of real optical beams is presented, and the mechanisms responsible for Raman beam cleanup and mode confinement are qualitatively discussed and modeled by computer simulations.