Nikolai Lobachevsky's father died when he was 7 years old. He attended Kazan Gymnasium, financed by government scholarships, from 1802 to 1807. Then he entered Kazan University. His original intention was to study medicine but he changed to study a broad scientific course involving mathematics and physics. Lobachevsky received a Master's Degree in physics and mathematics in 1811. In 1814 he was appointed to a lectureship, in 1816 he became an extraordinary professor, and in 1822 he was appointed as a full professor. He taught a wide range of topics including mathematics, physics and astronomy. His lectures were detailed and clear, so that they could be understood even by poorly prepared students.

He was soon appointed to important positions within the university, such as the dean of the Mathematics and Physics Department between 1820 and 1825, and head librarian from 1825 to 1835. He also served as Head of the Observatory and was clearly strongly influencing policy within the University. In 1827, he became rector of Kazan University, a post he was to hold for the next 19 years.

Despite this heavy administrative load, Lobachevsky continued to teach a variety of different topics such as mechanics, hydrodynamics, integration, differential equations, the calculus of variations, and mathematical physics. He even found time to give lectures on physics to the general public during the years 1838 to 1840, but the heavy work-load was to eventually take its toll on his health.

After Lobachevsky retired in 1846, his health rapidly deteriorated. His great mathematical achievements were not recognized in his lifetime, and he died without having any notion of the fame and importance that his work would achieve.

Since Euclid's axiomatic formulation of geometry, mathematicians had been trying to prove his fifth postulate as a theorem deduced from the other 4 axioms. Instead of trying to do this, Lobachevsky studied geometry in which the fifth postulate does not necessarily hold. He categorized euclidean as a special case of this more general geometry. In 1837, Lobachevsky published a summary of his new geometry.

In 1834, Lobachevsky also found a method for the approximation of the roots of algebraic equations. This method of numerical solution of algebraic equations is today a particularly suitable for methods for using computers to solve such problems.