On his mother's side Gvishiani is a grandson of Alexei Kosygin, Chairman of the USSR Council of Ministers in 1964–1980.

His main result was the proof of the Gelfand-Graev-Gvishiani formula for the characters of discrete series representations for the group of matrices with elements from a local non-Archimedean field (1973–74).

During his university studies he was influenced by prominent Soviet scientists – I. Gelfand, М. Keldysh, V. Chelomey, V. Sadovnichy and V. Maslov (presently academicians), M. Graev, А. Kirillov.

In 1991–2005 Gvishiani established and headed the Center for geophysical data studies and network technologies of IPE RAS.

In 2005 he was elected the director of the Geophysical Center of the Russian Academy of Sciences (GC RAS) and remained in that position till 2018.

He is also a member of the RAS Scientific Council on the complex problems of Eurasian economic integration, modernization, competitiveness and sustainable development and Fellow of the Russian Geographical Society.

Gvishiani made an outstanding contribution to the development of cooperation between Russian Academy of Sciences and the International Institute for Applied Systems Analysis (IIASA)/ as well as to promotion of cooperation between Russian scientists and the international scientific community.

Academician Gvishiani has been active in the field of international scientific cooperation: Vice President of the European Mediterranean seismological center (EMSC/CSEM) (EMSC/CSEM) (1996–2003); principal researcher of the joint project of RAS, RFBR and Scientific Research Center of France (CNRS) "Identification of areas of earthquake possible occurrence in regions of moderate seismicity" (PICS) (1980–1986, 1998–2001) and "Monitoring of basalt volcanoes with the use of artificial intelligence methods" (PICS) (2000–2006); delegate of RAS in the International Union of Geodesy and Geophysics (IUGG) (2007–2012) and CODATA (since 2005).

Over the years Gvishiani actively participates in the work of the International Council for Science (ICSU), holding various positions: Chairman of the coordinating Committee “Data Centers and Data Exchange” and Bureau member of the International Program “Lithosphere” (1985–1995); head of a number of CODATA WGs; member of the ICSU Strategic Committee on Information and Data (SCID) (2007–2008); Liaison officer of IUGG in CODATA (ICSU) (2011–2019).