The theory of features of differentiable maps - a rapidly developing area of modern mathematics , is a grand generalization of the study of functions to a maximum and minimum and has numerous applications in mathematics. natural science and technology (the so-called bifurcation and catastrophe theories). Chapters of the book are devoted to the theory of stability of smooth maps, critical points of smooth functions, features of caustics and wave fronts in geometric optics. The book is the first part of a large monograph conceived by the authors. The second part will outline the algebraic and topological aspects of the theory. The book is designed for mathematicians - from second-year students to researchers, as well as all consumers of feature theory in mechanics, physics, engineering and other sciences.