The mathematical description of catastrophes - abrupt changes that occur in the form of a sudden response of the system to a smooth change in external conditions, is given by the theories of singularities and bifurcations. Their applications to specific problems in various fields of science have caused much controversy. The book talks about what catastrophe theory is and why it causes such controversy. The results of mathematical theories of singularities and bifurcations are presented. The new edition is supplemented by a review of recent achievements in the theory of perestroika, a bibliography and a problem book. Designed for researchers, teachers, students and anyone interested in modern mathematics.