Lagrange was born at Turin in Italy of French ancestry. At school he was at first more interested in the classics until he read an essay by the astronomer Edmond Halley on the calculus, which converted him to mathematics. By the age of 18 he was teaching at the Artillery School in Turin, eventually becoming professor of mathematics there. He also began a discussion group, which evolved to form the Turin Academy of Sciences in 1758.

By 1760 Lagrange's reputation as one of Europe's greatest mathematicians was established; he received a prize from the Paris Academy of Sciences in 1764 for an essay on the libration of the Moon. On the invitation of Frederick the Great he succeeded Leonhard Euler as mathematical director of the Berlin Academy in 1766, serving there until 1787 when, following Frederick's death, he moved to Paris and a post in the Academy of Sciences.

Lagrange produced his greatest work, Mécanique analytique (1788; Analytical Mechanics), in Paris. This summarized the research in mechanics since Isaac Newton, based on Lagrange's own calculus of variations, and finally placed the mechanical theory of solids and fluids on a rigorous and analytical foundation. The book broke away from Euclidean tradition, and Lagrange commented in it that “one cannot find any diagrams in this work.”

Lagrange also made many contributions to astronomy and number theory. His work on the theory of equations helped Niels Abel in his later development of group theory. In astronomy he found a special solution to the three-body problem showing that asteroids will tend to oscillate around a certain point now called the Lagrangian point. Much of the credit for the revolutionary introduction of the metric system is also due to him.

Despite Lagrange's being technically a foreigner and also a friend of aristocrats he was regarded with respect during the French Revolution, which broke out in 1789, and was made president of the commission for metrication in 1793. He founded the mathematics department in the Ecole Normale in 1795 and later in the Ecole Polytechnique in 1797. Napoleon honored him, making him a count and a senator.