Application of high-order shock-capturing schemes to numerical simulation of problems in supersonic aerodynamics is considered. Euler and Navier-Stokes solvers based on employing modern weighted essentially non-oscillatory (WENO) schemes are described. A new WENO scheme, which has an advantage when applied in general curvilinear coordinates, is constructed. A number of examples of numerical simulations of 2D and 3D shock-dominated flows with high-order schemes are given. They include shock wave propagation and diffraction, shock/shock and shock/boundary layer interaction, development of hydrodynamic instability waves in high-speed free shear flows, and the structure of imperfectly expanded supersonic jets. For some problems, high-quality numerical schlieren visualizations and interferograms are compared with experimental patterns. We demonstrate that high-order WENO schemes is a powerful tool for simulation of compressible fluid flow. They can be considered as a very promising candidate for DNS and LES of turbulent supersonic flows.