The theory of singularities of differentiable maps is a rapidly developing area of contemporary mathematics, being a grandiose generalisation of the study of functions at maxima and minima and having numerous applications in mathematics, the natural sciences and technology (as in the so-called theory of bifurcations and catastrophes). Chapters of the book are devoted to the theory and stability of smooth maps, critical points of smooth functions, and the singularities of caustics and wave fronts in geometrical optics. The book is the first volume of a large monograph envisaged by the authors. In the second volume will be placed the algebro-topological aspects of the theory. The book is intended for mathematicians, from second year students up to research workers, and also for all users of the theory of singularities in mechanics, physics, technology and other sciences.