The results of numerous experiments on the search for quarks and other fundamental particles are reviewed. In the first section, various composite models of elementary particles are described: the Sakata model, the quark model, and models with two or three fundamental supermultiplets. The second section deals with the experiments on the search for quarks, polyquarks and fundamental particles with integral charges which have been carried out with accelerators. Special attention is given to the most recent highest-precision results obtained at IHEP (Serpukhov) and at CERN. The results of the accelerator experiments are analyzed by means of the phase-space model and a phenomenological model which takes into account the basic features of the dynamics of the strong interactions. No quarks of mass $\lesssim5$ GeV have been detected in the accelerator experiments, and the upper limits on the possible production cross sections of these particles have been estimated to be 10$^{-37}$--10$^{-39}$ cm$^2$. In the third section, the results of searches for quarks in cosmic rays and searches by physico-chemical methods are systematized. Quarks have not been detected in these experiments, and works reporting the observation of fractionally charged particles have proved to be in error. Various theoretical models of quark production are discussed in the fourth section. The review of the literature goes up to September 1972.