He was born at Beaune, Côte-d'Or. He was first educated at the college of the Oratorians at Beaune, and then in their college at Lyon - where, at sixteen, the year after he had been learning physics, he was made a teacher of it. Returning to Beaune for a vacation, he made, on a large scale, a plan of the town, inventing the methods of observation and constructing the necessary instruments; the plan was presented to the town, and preserved in their library. An officer of engineers seeing it wrote to recommend Monge to the commandant of the military school at Mézières, and he was received as a draftsman and pupil in the practical school attached to that institution; the school itself was of too aristocratic a character to allow his admission to it. His manual skill was duly appreciated: "I was a thousand times tempted," he said long afterwards, "to tear up my drawings in disgust at the esteem in which they were held, as if I had been good for nothing better."

An opportunity, however, presented itself: being required to work out from data supplied to him the defilading of a proposed fortress (an operation then only performed by a long arithmetical process), Monge, substituting for this a geometrical method, obtained the result so quickly that the commandant at first refused to receive it - the time necessary for the work had not been taken; but upon examination the value of the discovery was recognized, and the method was adopted. And Monge, continuing his researches, arrived at that general method of the application of geometry to the arts of construction which is now called descriptive geometry.

But such was the system in France before the Revolution that the officers instructed in the method were strictly forbidden to communicate it even to those engaged in other branches of the public service; and it was not until many years afterwards that an account of it was published.

In 1768 Monge became professor of mathematics, and in 1771 professor of physics, at Mézières; in 1778 he married Mme Horbon, a young widow whom he had previously defended in a very spirited manner from an unfounded charge; in 1780 he became a member of the Académie; his intimate friendship with C.L. Berthollet began at this time. In 1783, quitting Mézières, he was, on the death of É. Bézout, appointed examiner of naval candidates. Although pressed by the minister to prepare for them a complete course of mathematics, he declined to do so, on the ground that it would deprive Mme Bézout of her only income, from the sale of the works of her late husband; he wrote, however (1786), his Traité élémentaire de la statique.

Monge contributed (1770 – 1790) to the Memoirs of the Academy of Turin, the Mémoires des savantes étrangers of the Academy of Paris, the Mémoires of the same Academy, and the Annales de chimie, various mathematical and physical papers. Among these may be noticed the memoir "Sur la théorie des déblais et des remblais" (Mém. de l’acad. de Paris, 1781), which, while giving a remarkably elegant investigation in regard to the problem of earth-work referred to in the title, establishes in connection with it his capital discovery of the curves of curvature of a surface. Leonhard Euler, in his paper on curvature in the Berlin Memoirs for 1760, had considered, not the normals of the surface, but the normals of the plane sections through a particular normal, so that the question of the intersection of successive normals of the surface had never presented itself to him. Monge's memoir just referred to gives the ordinary differential equation of the curves of curvature, and establishes the general theory in a very satisfactory manner; but the application to the interesting particular case of the ellipsoid was first made by him in a later paper in 1795. (Monge's 1781 memoir is also the earliest known anticipation of Linear Programming type of problems, in particular of the transportation problem. Related to that, the Monge soil-transport problem leads to a weak topology definition of a distance between distributions rediscovered many times since by such as L.V. Kantorovich, P. Levy, L.N. Wasserstein, and others; and bearing their names in various combinations in various contexts.) A memoir in the volume for 1783 relates to the production of water by the combustion of hydrogen; but Monge's results had been anticipated by Henry Cavendish.

In 1792, on the creation by the Legislative Assembly of an executive council, Monge accepted the office of minister of the marine, and held this office from 10 August 1792 to 10 April 1793. When the Committee of Public Safety made an appeal to the savants to assist in producing the materiel required for the defence of the republic, he applied himself wholly to these operations, and distinguished himself by his indefatigable activity therein; he wrote at this time his Description de l'art de fabriquer les canons, and his Avis aux ouvriers en fer sur la fabrication de l'acier.

He took a very active part in the measures for the establishment of the normal school (which existed only during the first four months of the year 1795), and of the school for public works, afterwards the École Polytechnique, and was at each of them professor for descriptive geometry; his methods in that science were first published in the form in which the shorthand writers took down his lessons given at the normal school in 1795, and again in 1798 — 1799.

In 1796 Monge was sent into Italy with C.L. Berthollet and some artists to receive the pictures and statues levied from several Italian towns, and made there the acquaintance of General Bonaparte. Two years later he was sent to Rome on a mission which ended in the establishment, under A. Masséna, of the short lived Roman Republic; and he thence joined the expedition to Egypt, taking part with his friend Berthollet as well in various operations of the war as in the scientific labours of the Institut d'Égypte and Egyptian Institute of Sciences and Arts; they accompanied Bonaparte to Syria, and returned with him in 1798 to France. Monge was appointed president of the Egyptian commission, and he resumed his connection with the École Polytechnique. His later mathematical papers are published (1794 – 1816) in the Journal and the Correspondence of the École Polytechnique. On the formation of the Sénat conservateur he was appointed a member of that body, with an ample provision and the title of count of Pelusium (Comte de Péluse), and became the Sénat conservateur's president in 1806 - 07; but on the fall of Napoleon he was deprived of all his honours, and even excluded from the list of members of the reconstituted Institute.

Monge died at Paris on 28 July 1818 and was interred in Le Père Lachaise Cemetery, in Paris, in a mausoleum. He was later transferred to the Panthéon. A statue portraying him was erected in Beaune in 1849.