The main methods and results of classification of features of differentiable maps and functions with respect to different equivalence relations . are presented. Numerous applications of feature theory to differential geometry and mechanics are given: features of evolution, pores, rearrangement of wave fronts, maximum functions, etc. are described. An overview of the theory of boundary features of important subsets in functional spaces, such as sets of elliptic and hyperbolic polynomials, non-oscillational (Chebyshevian) systems, sets of functions that are fundamental solutions of differential equations is given. the beginning of the classification of these features. Numerous applications of the theory of monodromy are given: the theory of lacunae of hyperbolic operators, multidimensional generalizations of Newton's theorem on non-quadrability, hypergeometric functions and representations of Hecke algebras. A computer program is described that lists the decays of the features of real functions. Most of the results in the monographic literature are published for the first time.