This tutorial covers the basic asymptotic methods used in theoretical and deformable solid mechanics. Particular attention is paid to the mechanics of thin-walled structures. The presentation is illustrated by a large number of examples and problems that boil down to solving algebraic, transcendental, as well as ordinary differential equations. Along with regularly perturbed equations, solutions of singularly perturbed systems of equations, linear and nonlinear boundary value problems for eigenvalues are given.