The book systematically outlines the theoretical foundations of mathematical methods used in the analysis of risk situations. The main focus is on methods of analysis of insurance risks. Along with the material traditionally presented in the framework of lectures on risk theory and insurance mathematics, the book includes some sections containing the latest results. For students and graduate students studying in mathematical and economic-mathematical specialties (mathematics, applied mathematics, actuarial mathematics, financial mathematics, insurance business) . The book can be used by actuaries and analysts working in insurance and financial companies, as well as specialists in the field of reliability theory and other researchers whose activities are related to risk assessment and analysis of various risk situations. Approved by the Educational and Methodological Council for Applied Mathematics and Computer Science of UMO on classical university education as a teaching aid for university students , students in the specialty 010200 "Applied mathematics and computer science" and in the direction 510200 "Applied mathematics and computer science" .