Many shell supernova remnants are now known to radiate synchrotron X-rays. Several objects have also been detected in TeV gamma rays. Nonthermal X-rays and gamma rays can be produced in shell remnants by extremely energetic ions and electrons due to decay of π0 mesons produced in inelastic collisions between ions and thermal gas, or by electron synchrotron, bremsstrahlung, or inverse-Compton radiation. Thus observations at X-ray and gamma-ray wavelengths constrain the process of particle acceleration to high energies in the shock waves of supernova remnants. This review examines the relevant characteristics of Type Ia and core-collapse supernovae, the dynamics of their evolution through the Sedov blast-wave phase, the basic physics of diffusive shock acceleration, and the physics of the relevant radiative processes. It also reviews the current status of observations of shell remnants from X-rays to TeV gamma rays, and summarizes what we can learn about particle acceleration.