The Method of Intrinsic Scaling -- Weak Solutions and a Priori Estimates -- The Geometric Setting and an Alternative -- Towards the Hölder Continuity -- Some Applications -- Immiscible Fluids and Chemotaxis -- Flows in Porous Media: The Variable Exponent Case -- Phase Transitions: The Doubly Singular Stefan Problem.

This set of lectures, which had its origin in a mini course delivered at the Summer Program of IMPA (Rio de Janeiro), is an introduction to intrinsic scaling, a powerful method in the analysis of degenerate and singular PDEs. In the first part, the theory is presented from scratch for the model case of the degenerate p-Laplace equation. This approach brings to light what is really essential in the method, leaving aside technical refinements needed to deal with more general equations, and is entirely self-contained. The second part deals with three applications of the theory to relevant models arising from flows in porous media and phase transitions. The aim is to convince the reader of the strength of the method as a systematic approach to regularity for this important class of equations.