Progress in the search for a unified theory of elementary particles is reviewed. The supersymmetrical Kaluza-Klein theories are described: 11-, 10-, and 6-dimensional models of supergravity. The methods of spontaneous compactification, with whose help the four-dimensional theories are obtained, are described. The properties of the massless sector—zero modes in the Kaluza--Klein theories--and the question of the stability of vacuum solutions are discussed. An important criterion for the selection of a self-consistent theory is the absence of anomalies. The basic formulas for multidimensional chiral and gravitational anomalies are presented. The mechanism of the cancellation of the anomaly for Green and Schwarz's 10-dimensional effective field theory of superstrings with the gauge groups SO(32) and E$_8\times$ E$_8$ is described. The basic concepts and the results of the theory of superstrings are presented. This theory has no divergences and is at the present time a very attractive candidate for a unified theory of elementary particles.

@article{Aref’eva:1985,
	author = {I. Ya. Aref’eva and I. V. Volovich},
	title = {Supersymmetry: Kaluza-Klein theory, anomalies, and superstrings},
	publisher = {Physics-Uspekhi},
	year = {1985},
	journal = {Phys. Usp.},
	volume = {28},
	number = {8},
	pages = {694-708},
	url = {https://ufn.ru/en/articles/1985/8/c/},
	doi = {10.1070/PU1985v028n08ABEH003884}
}