The MoUs signed in presence of President Pranab Mukherjee - who is on a five-day visit to Russia - involves cooperation in the areas of mathematics, physics, chemistry and computer sciences.
The agreement will ensure cooperation, coordination and competition for conducting basic and exploratory scientific research carried out by international scientific teams.
"It aims at developing bilateral cooperation in the field of basic and exploratory scientific research and research training," a statement said.
"It envisages providing support to joint scientific projects during 2015-17 in the areas of mathematics, computer and systems science, physics and space science, chemistry and material science, biology and life sciences, basic research for medicine, agricultural sciences, earth sciences and engineering," it said.
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While some MoUs focus on agreement between IIT Bombay and Russia's National Tomsek State University to promote cooperation and encourage direct linkages between their faculty, others involve collaboration with IIT Madras for academic and research partnership.
A similar agreement involves collaboration with IIT Delhi for academic and research cooperation in areas of mutual interests.
Indian Statistical Institutes signed a MoU with Higher School of Economics while another agreement was signed by Kolkata's Institution of Engineers and Russian Scientific and Engineering Associations to promote and enlarge the exchange of technical, scientific and professional knowledge in both the countries.