The dynamics of a quantum system which is directly coupled to classical degrees of freedom is investigated. The classical degrees of freedom are in turn coupled to a classical bath whose detailed dynamics is not of interest. The resulting quantum-classical evolution equations are dissipative as a result of coupling to the classical heat bath. The dissipative quantum-classical dynamics is used to study nonadiabatic chemical reactions and compute their rates. The reactive flux correlation formalism for the calculation of nonadiabatic rate constants is generalized to dissipative quantum-classical dynamics and implemented in terms of averages over surface-hopping Langevin trajectory segments. The results are illustrated for a simple quantum-classical two-state model. The techniques developed in this paper can be applied to complex classical environments encountered, for example, in proton and electron transfer processes in the condensed phase where local environmental degrees of freedom must be treated explicitly but the remainder of the environment can be treated simply as a heat bath. (C) 2003 American Institute of Physics.