Introduction to upwind finite-volume methods widely used in computational fluids dynamics (CFD) for the solution of high-speed inviscid and viscous compressible flows. Topics include: Brief review of conservation equations for compressible flows; Euler equations; Navier-Stokes equations; one- and two-dimensional forms; model equations. Mathematical properties of the Euler equations; primitive and conserved solution variables; eigensystem analysis; compatibility conditions; characteristic variables, Rankine-Hugoniot conditions and Riemann invariants; Riemann problem and exact solution. Godunov's method; hyperbolic flux evaluation and numerical flux functions; solution monotonicity; Godunov's theorem. Approximate Riemann solvers; Roe's method. Higher-order Godunov-type schemes; semi-discrete form; solution reconstruction including least-squares and Green-Gauss methods; slope limiting. Extension to multi-dimensional flows. Elliptic flux evaluation for viscous flows; diamond-path and average-gradient stencils; discrete-maximum principle. High-order methods; essentially non-oscillatory (ENO) schemes.