Perturbation methods underlie almost all applications of physical applied mathematics, for instance, convection in porous media, fluid mechanics, celestial mechanics, traffic flow, optics, shock waves, reaction-diffusions and nonlinear oscillations. The aims of the course are to give a systematic account of modern perturbation methods and to show how it can be applied to differential equations (initial value problems, boundary value problems, eigenvalue problems). It includes introduction, methods of regular perturbations, methods of strained coordinates for periodic solutions, methods of matched asymptotic expansions, WKB methods, methods of multiple scales.