The nonequilibrium boundary layer is considered as a binary mixture of atoms and molecules with finite dates of dissociation and recombination. To obtain accurate solutions to the partial differential equations for this type of flow without any necessary simplifying assumptions, an implicit finite-difference scheme is developed for solving these equations with a digital computer. Accurate solutions to the nonequilibrium boundary-layer equations have been obtained in a reasonable amount of computer time and are presented for a flat plate, cone, and hemisphere cylinder. The results show that the nonequilibrium boundary-layer temperature and composition can be considerably different from local equilibrium and frozen results. For a cone at 21,000 fps and 100,000 ft alt, the computations show that, at 60 ft from the tip, the flow has not reached equilibrium.