In the first part of this book, the authors give a systematic account of methods to construct asymptotic solutions to semilinear parabolic and hyperbolic equations with a small parameter. A special emphasis is made on analysis of singular solutions in the limit of the vanishing small parameter. The classes of solutions studied in this book include solitary waves, shock waves, periodic traveling waves, etc. This methodological part of the book is chiefly based on formal asymptotic expansions and formal singularity analyses, but an Appendix (written by another author, S. A. Vakulenko) presents some important results concerning rigorous mathematical substantiation of the formal asymptotic expansions. In the second part of the book, the authors apply the general techniques developed in the first part to a number of applied problems, including nonlinear reaction-diffusion problems and laminar flow of a viscous liquid around a plate.