One of the great problems of astrophysics is the unanswered question about the origin and mechanism of chromospheric and coronal heating. Just how these outer stellar envelopes are heated is of fundamental importance, since all stars have hot chromospheric and coronal shells where the temperature rises to millions of degrees, comparable to the temperatures in the stars' cores. Here for the first time is a comprehensive inventory of the proposed chromospheric and coronal heating theories. The proposed heating processes are critically compared, and the observational evidence for the various mechanisms is reviewed. This is essential reading for all those working in such fields as stellar activity, radio and XUV emission, rotation, and mass loss, for whom a detailed and consistent presentation of our knowledge of chromospheric and coronal heating mechanisms is urgently needed.