The focus is on the disclosure of the concepts of stability and controllability of dynamic systems from the standpoint of modern catastrophe theory. Mathematical methods of theories of features and bifurcations are given. Classification of catastrophe types is given. A generalization of the mathematical methods of catastrophe theory is made in relation to the problems of functioning of systems of various nature. The application of mathematical methods and models of catastrophes in various branches of knowledge related to the analysis and assessment of the evolution of the behavior of complex dynamic systems is considered.