The founding of the Institute of Physical Problems — the unique project initiated by the Soviet Government — was undertaken for the continuation of scientific work of Peter Leonidovich Kapitza, the Director of the Mond Laboratory at Cambridge University, in the Soviet Union. The present paper describes the history of the P.L. Kapitza Institute for Physical Problems of the Russian Academy of Sciences from its founding in 1934 to the present. The evolution of the Institute along with the whole country, of its structure, goals, and scope of the projects is presented. The most outstanding results obtained by the scientists of the Institute are outlined, along with the current status and directions of research.