Excitable media are extended nonequilibrium systems having uniform rest states that are linearly stable but susceptible to finite perturbations. Depending on the forms of these perturbations, a variety of wave patterns can be triggered; solitary waves, target like patterns, and spiral waves are a few examples. These media are naturally encountered in biological systems and are also generated by a class of chemical systems, the most familiar of which is the Belousov-Zhabotinsky reaction. The recent progress that has been made in understanding patterns in one and two space dimensions is reviewed. Special attention is given to theoretical aspects, but experiments and numerical simulations are described as well. On the theoretical side two basic approaches are described, singular perturbation theories and kinematical theories. The two approaches have different ranges of validity and address different questions, but also have an overlap range that allows for comparison. The availability of large fast computers made extensive explorations of parameter spaces possible. New regimes of dynamical behavior found in that way are described. Finally, an account is given of the significant experimental progress that has been made recently by exploiting spectrophotometric and digital imaging techniques and by using new reactor designs.