A mixed quantum-classical formulation of nonadiabatic molecular processes is outlined. Based on a recently introduced mapping formalism [Stock and Thoss, Phys. Rev. Lett. 78, 578 (1997)], the formulation employs a quantum-mechanically exact mapping of discrete electronic states onto continuous variables, thus describing the dynamics of both electronic and nuclear degrees of freedom by continuous variables. It is shown that the classical evaluation of the mapping formalism results in a self-consistent description of electronic and nuclear degrees of freedom, which treats both types of dynamical variables in a completely equivalent way. The applicability of the approach is thus solely determined by the validity of the classical approximation and does not rest on additional assumptions such as the ad hoc combination of classical and quantum-mechanical theories. The observation of unrestricted flow of zero-point energy in the electronic degrees of freedom indicates the limits of the classical approximation. However, it is shown that this problem can virtually be removed by restricting the classically accessible phase-space. Adopting a multidimensional model of the internal-conversion process in the benzene cation, it is demonstrated that the classical mapping approach is able to account for the branching of classical trajectories in the presence of multiple surface crossings. The classical simulations are found to match the exact quantum-mechanical reference calculations quite accurately. The virtues and limitations of various mixed quantum-classical descriptions are discussed by comparing the mapping approach to the classical-path, the classical electron-analog, and the surface-hopping formulation, respectively.