Harnacks Inequality for Degenerate and Singular Parabolic Equations Degenerate and singular parabolic equations have been the subject of extensive research for the last 25 years. Despite important achievements the issue of the Harnack inequality for non negative solutions to these equations both of p Laplacian and porous medium type while raised by several authors has remained basically open. Recently considerable progress has been made on this issue to the point that except for the singular sub critical range both for the p laplacian and the porous medium equations the theory is reasonably complete. It seemed therefore timely to trace a comprehensive Annotation The authors give a comprehensive treatment of the Harnack inequality for non negative solutions to p laplace and porous medium type equations both in the degenerate (p 2 or m 1) and in the singular range (1(p(2 or 0(m(1) starting from the notion of solution and building all the necessary technical tools