Marie-Sophie Germain was born on April 1, 1776, in Paris, France, in a house on rue St. Denis. According to most sources, her father, Ambroise-Franҫois, was a wealthy silk merchant. In 1789, he was elected as a representative of the bourgeoisie to the États-Généraux, which he saw change into the Constitutional Assembly. It is therefore assumed that Sophie witnessed many discussions between her father and his friends on politics and philosophy. After his political career, Ambroise-Franҫois may have become the director of a bank; at least, the family remained well-off enough to support Germain throughout her adult life. Marie-Sophie had one younger sister, named Angélique-Ambroise, and one older sister, named Marie-Madeline. Her mother was also named Marie-Madeline, and this plethora of Maries may have been the reason she went by Sophie. Germain's nephew Armand-Jacques Lherbette, Marie-Madeline's son, published some of Germain's work after she died.

When Germain was 13, the Bastille fell, and the revolutionary atmosphere of the city forced her to stay inside. For entertainment she turned to her father's library. Here she found J. E. Montucla's L'Histoire des Mathématiques, and his story of the death of Archimedes intrigued her. Germain decided that if geometry, which at that time referred to all of pure mathematics, could hold such fascination for Archimedes, it was a subject worthy of study. So she pored over every math book in her father's library, even teaching herself Latin and Greek so she could read works like those of Sir Isaac Newton and Leonhard Euler. She also enjoyed Traité d'Arithmétique by Étienne Bézout and Le Calcul Différential by Jacques Antoine-Joseph Cousin. Later, Cousin visited her in her house, encouraging her in her studies. Germain's parents did not at all approve of her sudden fascination with mathmatics, which was then thought inappropriate for a woman. When night came, they would deny her warm clothes and a fire for her bedroom to try to keep her from studying, but after they left she would take out candles, wrap herself in quilts and do math. 

In 1794, when Germain was 18, the École Polytechnique opened. As a woman, Germain was barred from attending, but the new system of education made the “lecture notes available to all who asked." The new method also required the students to “submit written observations." Germain obtained the lecture notes and began sending her work to Joseph Louis Lagrange, a faculty member. She used the name M. LeBlanc, “fearing,” as she later explained to Gauss, “the ridicule attached to a female scientist." When Lagrange saw the intelligence of M. LeBlanc, he requested a meeting, and thus Sophie was forced to disclose her true identity. Fortunately, Lagrange did not mind that Germain was a woman, and he became her mentor. He too visited her in her home, giving her moral support.

Germain first became interested in number theory in 1798 when Adrien-Marie Legendre published Essai sur la théorie des nombres. After studying the work, she opened correspondence with him on number theory, and later, elasticity. Legendre showed some of Germain's work in the Supplément to his second edition of the Théorie des Nombres, where he calls it trés ingéniuse. Germain's interest in number theory was renewed when she read Carl Friedrich Gauss' monumental work Disquisitiones Arithmeticae. After three years of working through the exercises and trying her own proofs for some of the theorems, she wrote, again under the pseudonym of M. LeBlanc, to the author himself, who was one year younger than she. The first letter, dated 21 November 1804, discussed Gauss' Disquisitiones and presented some of Germain's work on Fermat's Last Theorem. In the letter, Germain claimed to have proved the theorem for n = p – 1, where p is a prime of the form p = 8k + 7; however, her proof contained a weak assumption. Gauss' reply did not comment on Germain's proof.

Around 1807, the French were occupying the German town of Braunschweig, where Gauss lived. Germain, concerned that he might suffer the fate of Archimedes, wrote to General Pernety, a family friend, requesting that he ensure Gauss' safety. General Pernety sent a chief of a battalion to meet with Gauss personally to see that he was safe. As it turned out, Gauss was fine, but he was confused by the mention of Sophie's name. Three months after the incident, Germain disclosed her true identity to Gauss. Although Gauss thought well of Germain, his replies to her letters were often delayed, and he generally did not review her work. Eventually his interests turned away from number theory, and in 1809 the letters ceased. Despite the friendship of Germain and Gauss, they never met.

When Germain's correspondence with Gauss ceased, she took interest in a contest sponsored by the Paris Academy of Sciences concerning Ernst Chladni's experiments with vibrating metal plates. The object of the competition, as stated by the Academy, was “to give the mathematical theory of the vibration of an elastic surface and to compare the theory to experimental evidence." Lagrange's comment that a solution to the problem would require the invention of a new branch of analysis deterred all but two contestants, Denis Poisson and Germain. Then Poisson was elected to the Academy, thus becoming a judge instead of a contestant, and leaving Germain as the only entrant to the competition. In 1809 Germain began work. Legendre assisted by giving her equations, references, and current research. She submitted her paper early in the fall of 1811, and did not win the prize. The judging commission felt that “the true equations of the movement were not established,” even though “the experiments presented ingenious results.” Lagrange was able to use Germain's work to derive an equation that was “correct under special assumptions.” The contest was extended by two years, and Germain decided to try again for the prize. At first Legendre continued to offer support, but then he refused all help. Germain's anonymous 1813 submission was still littered with mathematical errors, especially involving double integrals, and it received only an honorable mention because “the fundamental base of the theory [of elastic surfaces] was not established." The contest was extended once more, and Germain began work on her third attempt. This time she consulted with Poisson. In 1814 he published his own work on elasticity, and did not acknowledge Germain's help. Germain submitted her third paper, “Recherches sur la théorie des surfaces élastique” under her own name, and on 8 January 1816 she became the first woman to win a prize from the Paris Academy of Sciences. She did not appear at the ceremony to receive her award. Although Germain had at last been awarded the prix extraordinaire, the Academy was still not fully satisfied. Sophie had derived the correct differential equation, but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler, which led to incorrect boundary conditions.

After winning the Academy contest, she was still not able to attend its sessions because of the Academy's tradition of excluding women other than the wives of members. Seven years later this tradition was broken when she made friends with Joseph Fourier, a secretary of the Academy, and he got her tickets to the sessions.

Germain published her prize-winning essay at her own expense in 1821, mostly because she wanted to present her work in opposition to that of Poisson. In the essay she pointed out some of the errors in her method. In 1826 she submitted a revised version of her 1821 essay to the Academy. The revision included attempts to clarify her work by “introducing certain simplifying hypotheses." This put the Academy in an awkward position, as they felt the paper to be “inadequate and trivial,” but they did not want to “treat her as a professional colleague, as they would any man, by simply rejecting the work.” So Augustin-Louis Cauchy, who had been appointed to review her work, recommended she publish it, and she followed his advice. One further work of Germain's on elasticity was published posthumously in 1831: her “Memoir sur la courbure des surfaces.” She used the mean curvature in her research.

Germain's best work was in number theory, and her most significant contribution to number theory dealt with Fermat's Last Theorem. In 1815, after the elasticity contest, the Academy offered a prize for a proof of Fermat's Last Theorem. It reawakened Germain's interest in number theory, and she wrote to Gauss again after ten years of no correspondence. In the letter, Germain said that number theory was her preferred field, and that it was in her mind all the time she was studying elasticity. She outlined a strategy for a general proof of Fermat's Last Theorem, including a proof for a special case. Germain's letter to Gauss contained the first substantial progress toward a proof in 200 years. She asked Gauss if her approach to the theorem was worth pursuing. Gauss never answered.

In addition to math, Germain studied philosophy and psychology. She wanted to classify facts and generalize them into laws that could form a system of psychology and sociology, which were then just coming into existence. Her philosophy was highly praised by Auguste Comte. Two of her philosophical works, Pensées diverses and Considérations générales sur l'état des sciences et des letteres aux différentes epoques de leur culture, were published, both posthumously. This was due in part to the efforts of Lherbette, her nephew, who collected her philosophical writings and published them. Pensées is a history of science and mathematics with Sophie's commentary. In Considérations, the work admired by Comte, Sophie argues that there are no differences between the sciences and the humanities.

In 1829 Germain learned she had breast cancer. Despite the pain, she continued to work. In 1831 Crelle's Journal published her paper on the curvature of elastic surfaces. She also published in Annales de chimie et de physique an examination of principles which led to the discovery of the laws of equilibrium and movement of elastic solids. On June 27 of 1831, she died in the house at 13 rue de Savoie.