This book is a reference on nonlinear second-order parabolic partial differential equations that are used as models to help solve a broad class of engineering and physical problems. The analytical ideas used are geometric as opposed to super-sub solution methods that have limited application in the physical world. Sturm Theory is the cornerstone of this type of analysis, which was developed almost two hundred years ago, but then lost or forgotten. Recently the theory was revived and used to produce dramatic mathematical results. At present, there are a number of books that promote the Sturm Theory for ordinary differential equations, but none of them are devoted to parabolic partial differential equations.