A new implicit finite-volume Single-Sweep ParabolizedNavier-Stokes (SSPNS) algorithm is developed. Theoretical analysis isfocused on the mathematic properties about the parabolized Navier-Stokes(PNS) Equations, especially on the treatment of streamwise pressuregradient. Then the original implicit time iterative Lower-Upper SymmetricGauss-Seidel (LU-SGS) method is successfully extended to integrate the PNSEquations in the streamwise direction. The hybrid upwind schemes, includingAdvection Upstream Splitting Method (AUSM) family schemes and Low-DiffusionFlux-Splitting (LDFSS) schemes, are used to compute the crossflow inviscidfluxes, while central schemes for the viscous fluxes. Four typical flows,i.e., supersonic flat plate flow, 15$^\circ$ ramp hypersonic flow,10$^\circ$ coneflows with different angles of attack, and side-compression hypersonic inletflows are calculated with the SSPNS codes. Numerical results agree well withthose obtained from NASA's UPS PNS codes and experimental results by Tracy,Holden, or Holland et al. All the numerical results indicate that SSPNS is ahighly efficient, highly accurate, and also highly robust algorithm forsteady supersonic/hypersonic flows without any large streamwise separation.Comparison to the traditional time-iterative Full Navier-Stokes (FNS) flowsolvers, the SSPNS codes show 1~2 order of magnitude of computationalspeed faster and at least 1 order of magnitude of memory storage saving.