The theory of active systems - a section of the theory of management of socio-economic systems, studying the properties of the mechanisms of their functioning due to the activity of the participants of the system. This work is an introduction to the theory of active systems and reflects the content of the annual course of lectures given by the authors and their colleagues for many years to students of the Moscow Institute of Physics and Technology and other universities. The theoretical emphasis of this course is due to the desire to give the reader a general idea of the research methods used in the study of mathematical models of socio-economic systems.