The theory of features of differentiable maps is a rapidly developing area of modern mathematics , It 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). The first part of the book is devoted to the theory of stability of smooth maps, critical points of smooth functions, features of caustics and wave fronts in geometric optics. The second part deals with families of complex hypersurfaces, asymptotics of integrals of multidimensional methods of stationary phase and pass, applications of algebraic geometry methods to the study of critical points of functions. For mathematicians - researchers, graduate students, students, as well as for specialists in the field of mechanics, physics, engineering and other sciences interested in the theory of the features of differentiable maps . The previous edition of the book was published in 2004.