This review is devoted to a new method in experimental physics--electron cooling, which opens the possibility of storing intense and highly monochromatic beams of heavy particles and carrying out a wide range of experiments with high luminosity and resolution. The method is based on the cooling of beams by an accompanying electron flux as the result of Coulomb collisions of the particles. In the first part of the review the theoretical basis of the method is briefly considered. The apparatus NAP-M is described with its electron cooling, and the results of successful experiments on cooling a proton beam are presented. In the second part the new possibilities opened by the use of electron cooling are discussed: storage of intense beams of antiprotons and achievement of proton-antiproton colliding beams, performance of experiments at the ultimate in low energies (with participation of antiparticles), storage of polarized antiprotons and other particles, production of antiatoms, storage of antideuterons, and experiments with ion beams.