This book surveys various mathematical approaches to perturbation theory and illustrates their applicability on a number of simple, physically motivated examples and exercises expressed in terms of differential equations. These methods simplify the original formulation, still giving under certain conditions the essential characteristics of analysed processes. The topics covered include: (i) tools to study regular and singular perturbations, (ii) renormalization, characteristics and multiple scale methods for one-dimensional time-dependent nonlinear waves, (iii) the Padé approximation and its application in mechanics, (iv) oscillators with negative Duffing type stiffness, and (v) differential equations with discontinuous nonlinearities. The importance of understanding the physical concepts behind the model under consideration is emphasized throughout the book.