A new forward/backward sweeping parabolized Navier-Stokes algorithm has been developed to compute efficiently supersonic/hypersonic flowfields with embedded separated regions. The algorithm splits the streamwise flux vector using the Steger-Warming method and employs multiple forward/backward sweeps of the flowfield to duplicate the results that would be obtained with the complete Navier-Stokes equations. The forward/backward sweeping of the flowfield significantly reduces the number of iterations required over previous iterative parabolized Navier-Stokes algorithms. Once a separated flow region is computed, the algorithm returns to the usual forward-space-marching mode until the next separated flow region is encountered. The algorithm has been successfully incorporated into NASA's parabolized Navier-Stokes UPS code. The new algorithm has been applied to three separated flow test cases consisting of flow over a compression ramp and two flows over a hollow-cylinder-flare geometry. The present numerical results are in excellent agreement with complete Navier-Stokes computations and experimental data.