This work will serve as an excellent first course in modern analysis. Key topics in nonlinear pde's as well as several fundamental tools and methods are presented; few prerequisites are required of the reader. Challenging exercises, examples, and illustrations help explain the rigorous analytic basis for the Navier-Stokes equations, mean curvature flow equations, and other important equations describing real phenomena. The main focus of the text is on showing how self-similar solutions are useful in studying the behavior of solutions of nonlinear partial differential equations, especially those of parabolic type. The exposition moves systematically from the basic to more sophisticated concepts, and in the final chapters recent developments and several open problems are presented. An extensive index is provided.Written for graduate students and researachers by one of Japan's leading analysts, this will be an excellent resource for self-study or classroom use.