Research in mathematics and mechanics has been conducted since Kazan University was founded in 1804. World-famous mathematicians have studied and worked at Kazan University: from 1827 to 1845 the president of Kazan University was Nicolay Lobachevsky – the Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry. English mathematician William Kingdon Clifford later called Lobachevsky the “Copernicus of Geometry” due to the revolutionary character of his work. Mathematics and mechanics research was initially conducted within the framework of the Department of Physical and Mathematical sciences, and later at the Faculty of Physics and Mathematics.

In 1960 the Faculty of Physics and Mathematics was divided into two units – the Faculty of Mechanics and Mathematics (currently the Nicolay Lobachevsky Institute of Mathematics and Mechanics) and the Faculty of Physics. In 1978 the Faculty of Computing Mathematics and Cybernetics (currently the Institute of Computer Mathematics and Information Technologies) was separated from the Faculty of Mechanics due to its rapid growth and development.

The Faculty is one of the leading educational and research centers of applied mathematics and computer science in Russia – the fastest-developing knowledge domain. The Institute of Computing Mathematics and Information Technologies trains experts in a wide range of specialties connected with the application of computing techniques in education, science, industry, economics and management.

The Department of Astronomy at KFU is the oldest in Russia, and professional astronomers have been trained here since 1810. Among the department's first stu￾dents were future acclaimed mathematician Nikolai Lobachevsky, and Ivan Simonov, who later took part in the round-the-world expedition of Fabian Gottlieb von Bellingshausen and Mikhail Lazarev that discovered Antarctica and many other islands.

Between 1833 and 1837, former rector of KFU Nikolai Lobachevsky decreed that an observatory be built in Kazan, which became one of the greatest in the world at the time. Over the course of the next few years a nine-inch refractor, meridian circle, heliometer and other astronomical instruments were installed, and regular observations began to be made at the observatory.

In the 1930s, a new department was created at the site – the Department of Geodesy. Teachers took part in geodesic and gravimetric surveys of the Volga Region and Ural Mountains. An astronomy and geodesy branch was then formed, growing to accommodate 30 people in 1931, and together with the astronomy branch began to train specialists in geodesy, gravimetry and cartography.

The Scientific School of Kazan is recognized both throughout Russia and abroad. Between 1960 and 2000, scientific research has been conducted in conjunction with astronomers from the Russian Academy of Sciences (RAS) Special Astrophysical Observatory, RAS Institute of Astronomy, RAS Space Research Institute, RAS Institute of Theoretical Astronomy, Sternberg Astronomical Institute of Moscow University (GAISh), Crimean Astrophysical Observatory, National Academy of Sciences of Ukraine Main Astronomical Observatory, Odessa National University, Radioastrophysics Observatory of the Latvian SSR Academy of Sciences, Astrophysics Observatory of the Estonian Academy of Sciences, Astronomical Institute of Utrecht University (Netherlands), Joint Institute for Laboratory Astrophysics (Boulder, USA), University of London (UK), Instituto de Astrofísica de Canarias (Spain), Max Planck Institute for Astrophysics (West Germany) and the US Space and Rocket Center of Alabama (USA). Results include material from unique observations made by the satellites TD-1A, BUSS and CHANDRA and the largest terrestrial telescopes (6m and 1m telescopes at the RAS Special Astrophysical Observatory and the 2.2m telescope at the Calar Alto Observatory in Italy); the program exchange for spectral processing to calculate theoretical spectra; and the publication of 22 collaborative articles in international and internal journals.

Nowadays KFU researchers supervise the V.P. Engelgardt Astronomical Observatory (AOE) and the North Caucasus Astronomical Station of KFU (AS KFU).

The scientific department was already in existence at the AOE before the astronomical observatory itself was opened in 1899, and at the time functioned as a meteoological station, so when the observatory was officially opened, it had already been in use for some time. Thus was formed the Department of Astronomy, which has stood the test of time and is still a part of the AOE today.

All scientific work at the AOE was concentrated in the Department of Astronomy – observations from the telescopes at the Astronomical Observatory, results from gravimetric expeditions to Novaya Zemlya and the processing of astronomical observations were all taken here. The astronomical observations here were second to none, and often surpassed those made at universities elsewhere across the globe. Suffice it to say, the AOE was always seen as being the pick of the scientific departments. More than once, researchers at the Engel`gardt Observatory were appointed head of Russian national observatories (such as Dmitri Martynov, Vladimir Krat and Aleksandr Yakobin), as well as became well-known scientists overseas (such as Tadeusz Banachiewicz). From its very inception, the AOE became a stand-out school of astronomy and began to create many top-quality specialists. The next big structural change came in 1931, with the creation of the Department of Astrophysics under the guidance of Dmitri Martynov.

Through the efforts of Konstantin Kostylev, the AOE Meteor Department was created in 1957, and meteors began to be observed by means of a photoelectric meteor patrol. The Lunar Department was then created by Anatoly Nefed`ev in 1966; a period when scientists across the world were engaged in intensive research into our only natural satellite. The department joined a network of groups across the world that studied the kinematics and dynamics of the moon. In 1970 the AOE Research and Development Group was formed under the guidance of Oleg Shornikov, the goal of which was to create new technology for observing and measuring, and also to refine and perfect existing apparatus.

In 1990 the Lunar Department was converted into the Department of Photographic Astrometry, and under the stewardship of Naufal Rizvanov dealt with lunar matters and with the creation and examination of photographic star catalogs.

Astronomy at Kazan became known across the world for lunar research, defining the positions and physics of stars, the study of the movement of celestial bodies and the dangers posed by meteors to astronauts. Not by chance have many celestial bodies have taken their names from Kazan scientists: comets ('Dubyago 1921 I' and 'Dubyago 1923 III' in honor of Aleksandr Dubyago), minor planets ('Dubyago', 'Emelyanenko', 'Krat', 'Lobachevsky' and 'Martynov'), lunar craters ('Banachiewicz', 'Bel`kovich' in honor of Igor Bel`kovich, 'Grachev', 'Dubyago' in honor of Dmitri Dubyago, 'Koval`skij', 'Krasnov', 'Littrow' and 'Lobachevskiy') and the 'Gusev' crater on Mars, in honor of Matvei Gusev.

Nowadays the department conducts research in astrometry, selenodesy and celestial mechanics, as well as placing much value on astrophysical research. Many of the department's scientists are members of the International Astronomical Union, and a further 5 are members of the European Astronomical Society.

The Higher Institute for Information Technology and Information Systems (ITIS) is one of the youngest and the most innovative departments of Kazan Federal University. It was founded in 2011 on the initiative of the Ministry of Information and Communication of the Republic of Tatarstan, Kazan Federal University and IT-Park. From the start, the Higher Institute for ITIS was supported by some of the largest IT-companies in the world, such as Microsoft, Fujitsu, Oracle, IBM, HP and Cisco, among others.

We have built a strong educational system of teaching applied disciplines. Most importantly, our students have access to the research laboratories of world-leading IT companies based at the university, such as Microsoft, Fujitsu and Samsung. In October 2014 Cisco Systems opened the Cisco Innovation Center at the Institute – the company's first research center in Russia to be based in a University.