The aim of the present short review is the exposition of the fundamental results obtained by Academician I. N. Vekua (1907-1977) in the theory of shells. The review deals with questions of constructing different versions of shell theory, questions of the infinitesimal bending of a surface of positive curvature and equilibrium membrane states of stress of a convex shell, and also the statically determinable problems and questions of existence of a neutral surface of the shell, i.e. the questions which Vekua investigated in different periods of his versatile scientific actively.
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