The drift-diffusion equations of semiconductor physics, allowing for field-dependent drift velocities, are analysed by the method of matched asymptotic expansions for one-dimensional PN and PNPN forward-biased structures. The analysis is relevant to describing the structure of the solutions to the drift-diffusion equations for large electric fields when drift velocity saturation effects become significant. In this high-field limit, the boundary layer structure for the solutions to the drift-diffusion equations is seen to differ substantially from that near equilibrium. In particular, boundary layers for the carrier concentrations can occur near the contacts. The asymptotic solutions and the current-voltage relations, constructed in the high-field limit, are found to agree well with direct numerical solutions to the drift-diffusion equations. © 1992 Oxford University Press.