[Authors] Kuznetsov, N. G.; Nazarov, A. I. [Title] Sharp constants in Poincar'e, Steklov and related inequalities (a survey) [AMS Subj-class] 35A23 Inequalities involving derivatives and differential and integral operators, inequalities for integrals [Abstract] The 150th anniversary of the birth of the outstanding Russian mathematician Vladimir Andreevich Steklov falls on January 9, 2014. In this paper we describe advances in one of several areas influenced by Steklov's work. [Keywords] Steklov inequality, Poincar'e inequality, Sobolev inequality [Comments] LaTeX, English, 24 pp. [Contact e-mail] al.il.nazarov@gmail.com