Physicists are becoming more interested in solving the equations of quantum theory without using methods requiring expansion in powers of the interaction constant. The perturbation-theory method, which gave results that agree splendidly with experiment in quantum electrodynamics, turned to be inapplicable in strong-interaction theory. One of the methods in which a radical attempt is made to go beyond the framework of perturbation theory, is the method of functional integration in quantum theory, first proposed by Feynman. The present review, which is devoted to this method, introduces in lucid fashion the concepts of functional integrations and then explains some applications of this method in quantum field theory. Much attention is paid to the use of functional integrals in infrared and high-energy asymptotic relations in field theory. The review does not claim an exposition of the mathematical difficulties connected with the concept of functional integral, and focuses attention to certain successes in its use in quantum physics.