The active area of research investigation during the past sixty years has been the study of solitons and related nonlinear real world phenomena that cannot be explained with the linear evolution equations. A Lagrangian based approach has been derived to study the properties of the nonlinear evolution equations. This book will provide the background of fundamental ideas and classical Lagrangian approach to understand nonlinear real-world wave phenomena, soliton dynamics, symmetry, supersymmetric realization and some important methods for solving the nonlinear equations up to the level of present-day active research on these and related topics. I hope this will stimulate future research on understanding the nonlinear problems and may be a useful source book for researchers, graduate students enrolled in M.S and Ph.D degree programs. Considering the multidisciplinary nature and important applications almost all universities offer a high-level graduate course devoted to frontier topics like nonlinear wave theory, integrable systems, solitons, symmetry and supersymmetry. It will be a useful text for graduate and senior-level courses dealing with above topics.