This book is an introduction to the perturbation theory for linear and nonlinear waves in dispersive and dissipative media. The main focus is on the direct asymptotic method which is based on the asymptotic expansion of the solution in series of one or more small parameters and demanding finiteness of the perturbations; this results in slow variation of the main-order solution. The method, which does not depend on integrability of basic equations, is applied to quasi-harmonic and non-harmonic periodic waves, as well as to localized waves such as solitons, kinks, and autowaves. The basic theoretical ideas are illustrated by many physical examples throughout the book.Contents:Perturbed Oscillations and Waves: Introductory ExamplesPerturbation Method for Quasi-Harmonic WavesPerturbation Method for Non-Sinusoidal WavesNonlinear Waves of ModulationPerturbation Methods for Solitary Waves and FrontsPerturbed SolitonsInteraction and Ensembles of Solitons and KinksDissipative and Active Systems. AutowavesReadership: Graduate students and young researchers in nonlinear science, physicists and applied mathematicians.