Some developments and efforts in designing and analyzing shock capturing algorithms and related numerical methods in computational fluid dynamics are reviewed. The importance of numerical viscosity in shock capturing algorithms is analyzed; the convergence and stability of some shock capturing algorithms are presented; the role of shock capturing algorithms in a mathematical existence theory is exhibited, especially for the compressible Euler equations for gas dynamics in one dimension and in multi-dimensions with spherical symmetry. Applications of shock capturing ideas to the compressible Navier-Stokes equations are also discussed.