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Mathematical research in Georgia began immediately after the opening of Tbilisi State University (1918). The initiators of the research were the first Georgian scientists in mathematics A. Razmadze, N. Muskhelishvili, A. Kharadze and G. Nikoladze.

On October 8, 1933 a research institute of mathematics, physics and mechanics with N. Muskhelishvili as director was set up under Tbilisi State University.

On October 1, 1935, at the initiative of N. Muskhelishvili and his closest associates V. Kupradze and I. Vekua, the mathematics and mechanics section of the above-mentioned institute was transformed into a mathematical research institute under the auspices of the Georgian Branch of the USSR Academy of Sciences. The institute was included into the Georgian Academy of Sciences after the latter was founded in February, 1941. In 1944 the institute was granted the name of A. Razmadze.

Along with N. Muskhelishvili, V. Kupradze and I. Vekua, to the first generation of mathematical researchers of the institute belong A. Bitsadze, V. Chelidze, G. Chogoshvili, D. Dolidze, L. Gokieli, A. Gorgidze, A. Kalandia, I. Kartsivadze, A. Kharadze, B. Khvedelidze, D. Kveselava, G. Lomadze, L. Magnaradze, K. Marjanishvili, Sh. Mikeladze, A. Rukhadze, N. Vekua, A. Walfisz. Their fundamental results on the mathematical theory of elasticity [28, 48-56, 71, 72, 74, 75, 77, 80, 91, 97-100], singular integral equations [73, 76, 78, 79, 81, 82, 101-103], complex and real analysis [7, 8, 31-33, 93, 95, 96], differential equations and mathematical physics [2-5, 44-47, 70, 83, 90, 92, 94], topology [10-14], the theory of numbers [104-108] and computational mathematics [60-68] won the institute a high scientific reputation worldwide.

At various times, with institute collaborated the outstanding foreign scientists P. S. Aleksandrov, S. Bergman, S. N. Bernstein, M. V. Keldysh, M. A. Lavrentyev, V. I. Smirnov and S. L. Sobolev.

The first director of the institute was V. Kupradze (1935-1941). Throughout 1941-1976 the institute almost unintermittently was headed by N. Muskhelishvili. From September, 1951 to April, 1952 the acting director of the institute was V. Chelidze. In 1976-1989 the institute was directed by N. Vekua, and in 1989-2006 by I. Kiguradze. From June, 2006 to April, 2008 the acting director of the institute was N. Partsvania. Since April, 2008 N. Partsvania is a director of the institute.

The institute's deputy directors for scientific research were I. Vekua (1940-1941), A. Gorgidze (1941-1954), G. Manjavidze (1954-1977) and T. Burchuladze (1977-1989),  V. Kokilashvili (1989-2006). Since June 2006, the deputy director of the institute is R. Gachechiladze.

At various times, the research departments of the institute were headed by R. Bantsuri, S. Bergman, S. Bernstein, V. Chelidze, G. Chogoshvili, T. Gegelia, A. Kalandia, R. Kapanadze, A. Kharadze, E. Khmaladze, B. Khvedelidze, V. Kupradze, L. Magnaradze, G. Mania, M. Mikeladze, Sh. Mikeladze, N. Muskhelishvili, T. Shervashidze, O. Tsereteli, I. Vekua, N. Vekua, A. Walfisz, and the scientific workers of the institute were  R. Abdulaev, R. Chichinadze, A. Dzhishkariani, J. Gvazava, G. Janashia, T. Kandelaki, G. Khuskivadze, O. Maisaia, E. Obolashvili, D. Pataraia,  G. Tevzadze, A. Tsitskishvili.

Presently, the institute has 9 scientific departments:

Since 1937 the institute has been publishing the scientific journal "Proceedings of A. Razmadze Mathematical Institute" (from 2016 is published by ELSEVIER as Transactions of A. Razmadze Mathematical Institute)

The institute is the founder of two international scientific journals: "Georgian Mathematical Journal" and "Memoirs on Differential Equations and Mathematical Physics".

As from 1997, the Symposium on Differential Equations and Mathematical Physics (DEMPh) is annually held at the institute