The quantum-classical Liouville formulation gives a quantum-mechanical density-matrix description of the "quantum" particles of a problem (e.g., the electrons) and a classical phase-space-density description of the "classical" particles (e.g., the nuclei). In order to employ this formulation to describe multidimensional nonadiabatic processes in complex molecular systems, this work is concerned with an efficient Monte Carlo implementation of the quantum-classical Liouville equation. Although an exact stochastic realization of this equation is in principle available, in practice one has to cope with two major complications: (i) The representation of nonlocal phase-space operators in terms of local classical trajectories and (ii) the convergence of the Monte Carlo sampling which is cumbersome due to complex-valued trajectories with rapidly oscillating phases. Several strategies to cope with these problems are discussed, including various approximations to determine the momentum shift associated with a nonadiabatic transition, the on-the-fly generation of new trajectories at curve-crossings, and the localization of trajectories after irreversible electronic transitions. Employing several multidimensional model systems describing ultrafast photoinduced electron transfer and internal conversion, detailed numerical studies are performed which are compared to exact quantum calculations as well as to the "fewest-switches" surface-hopping method. In all cases under consideration, the Liouville calculations are in good agreement with the quantum reference. In particular, the approach is shown to provide a correct quantum-classical description of the electronic coherence. (C) 2001 American Institute of Physics.