The Krasovskii Institute of Mathematics and Mechanics is a federal government research institution that conducts basic and applied research in mathematics, mechanics, and computer science and puts the results into practical use. The main research directions of the institute include:

• development of the mathematical theory of control processes;
• studies in algebra, topology, analysis, function theory, differential equations, and mechanics;
• research in the theory and methods of mathematical programming and pattern recognition, mathematical physics and continuum mechanics, and ill-posed and inverse problems;
• development of mathematical modeling methods, algorithms, and software for problems of control, optimization, pattern recognition, mathematical physics, and mechanics;
• development of high-performance computing facilities and operational software for parallel and distributed computing systems, development of a regional network of scientific telecommunications and computer sharing, and creation of telecommunications support software.

The institute was established in 1956 as the Sverdlovsk Division of the Steklov Institute of Mathematics.  Its building at 16 S. Kovalevskaya Street was laid in 1957 and put into operation in 1961. In 1970, the division was transformed into the Institute of Mathematics and Mechanics of the Academy of Sciences of the USSR.

After Academician N.N. Krasovskii's death in 2012, the institute was given his name. The institute works under the scientific and methodological leadership of the Division of Mathematical Sciences of the RAS and the Division of Mechanics and Control Processes of the RAS.

Throughout its history, the institute was headed by Academician S.B. Stechkin, Academician N.N. Krasovskii, Academician A.B. Kurzhanskii, Prof. Dr. V.D. Batukhtin, Academician Yu.S. Osipov, Academician A.F. Sidorov, and Academician V.I. Berdyshev. Currently, Prof. Dr. N.Yu. Lukoyanov is Director of the institute.

The Krasovskii Institute of Mathematics and Mechanics is the publisher of the quarterly journal Trudy Instituta Matematiki i Mekhaniki UrO RAN (its English translation is published as supplement issues of the Proceedings of the Steklov Institute of Mathematics) and the semi-annual online Ural Mathematical Journal.