Henri grew up in a home in which music was very important with all the children playing music. Despite Henri's father being a mathematician, he did not try to influence his children towards a career in mathematics. Although always prepared to answer Henri's questions about mathematics as he was growing up, he never emphasised the subject more than others. Despite this Henri always knew from a very young age that he would be a mathematician.
After completing his school education Henri studied at the École Normale Supérieure in Paris and soon became friendly with André Weil who was one year ahead. Among Henri Cartan's teachers at the École Normale were Gaston Julia and his father Élie Cartan. The students at the École Normale also had to attend general courses at the Sorbonne, so Henri studied there too. His doctoral studies were supervised by Paul Montel, whose research interests were the theory of analytic functions of a complex variable, and Cartan received his Docteur ès Sciences mathématiques in 1928. After being awarded his doctorate, Cartan taught at the Lycée Caen from 1928 to 1929, then at the University of Lille from 1929 to 1931.
It had been Cartan's friend André Weil who had suggested that he work on analytic functions of several complex variables and Weil told him about the work of Carathéodory. Cartan published Les transformations analytiques des domaines cerclés les uns dans les autres Ⓣ in 1930 and, since this paper contained generalisations of results proved by Heinrich Behnke, he was invited by Behnke to visit Germany in May 1931 and give a series of lectures at Münster in Westphalen where Behnke taught. While there he met Peter Thullen, Behnke's assistant, and they began a collaboration which resulted in the joint paper Zur Theorie der Singularitäten der Funktionen mehrerer Komplexen Veränderlichen Ⓣ published in Mathematische Annalen in 1932. Thullen (although he was not Jewish) left Germany after the Nazis came to power in 1933 so the collaboration came to an end. Cartan did visit Behnke in Münster in Westfalen for a second time before the start of World War II, accepting an invitation to go there in 1937.
A joint paper written by Henri and his father Élie Cartan, Les transformations des domaines cerclés bornés Ⓣ appeared in 1931. Most of the time the two mathematicians worked independently, but for this paper they were able to use Élie Cartan's expertise on Lie groups to tackle questions that Henri had been interested in. The year in which this joint paper appeared was the one when Cartan left his position in Lille and, starting in November 1931, he took up a post at the University of Strasbourg. He married Nicole Antoinette Weiss on 14 September 1935. They had two sons and three daughters: Jean, Francoise, Étienne, Mireille, and Suzanne.
A very important part of Cartan's mathematical life was taken up with Bourbaki. The first meeting of the group of mathematicians who called themselves "Bourbaki" took place on 14 January 1935. In  Cartan explains the background to Bourbaki:-
[A]fter the First World War, there were not so many scientists, I mean good scientists, in France, because most of them had been killed. We were the first generation after the war. Before us there was a vacuum, and it was necessary to make everything new. Some of my friends went abroad, notably to Germany, and observed what was being done there. This was the beginning of a mathematical renewal. It was due to such people as Weil, Chevalley, de Possel .... The same people, responding to André Weil's initiative, came together to form the Bourbaki group.Leray and Paul Dubreil attended the January 1935 meeting, but dropped out before membership of the group was finalised in July of 1935. In addition to Henri Cartan the founding members of Bourbaki at that July meeting were André Weil, Jean Dieudonné, Szolem Mandelbrojt, Claude Chevalley, René de Possel, and Jean Delsarte. Cartan described how the group operated :-
We often disagreed, we often had big arguments - but we remained good friends. For each subject, a "rédacteur" was appointed. Later, his rédaction was read aloud and thoroughly examined. The next "rédacteur" was given the appropriate instructions, and so on. For each chapter there could be up to nine rédactions. But in the end, everybody was fatigué - tired. And Dieudonné would say, "It is finished now. I shall write the last rédaction." Which he did. And eventually, although it seemed to be impossible to reach a complete agreement, there was an agreement. But it took time. It is perhaps not the best way in terms of teamwork, but that was the way we took.Cartan was teaching at the University of Strasbourg when World War II started :-
But in September 1939, the inhabitants of Strasbourg had to be evacuated. The university was displaced to Clermont-Ferrand, where I taught for a year before I was appointed professor at the Sorbonne in Paris, in November 1940 (in fact, I was to be in charge of the mathematics students at the École Normale). Throughout the war I was not permitted to go to my apartment in Strasbourg. One day, Behnke offered to try and retrieve some mathematical papers I had left there. He actually went to Strasbourg, but to no avail. He tried again and succeeded. He managed to get hold of some documents, which he left with the library of the University of Freiburg. In 1945, some members of the French Forces in Germany happened to find them there and returned them to me.It has to be remembered that the period of the war was one of great tragedy for the Cartan family. Henri's brother Louis was a member of the Resistance fighting in France against the occupying German forces. After his arrest in February 1943 the family received no further news but they feared the worst. Only in May 1945 did they learn that he had been beheaded by the Nazis in December 1943. Late in 1945, when World War II had ended, Cartan returned to the University of Strasbourg and taught there for a further two years. He renewed contacts with his German colleagues when he visited the Research Institute in Oberwolfach, in the Black Forrest, in November 1946 :-
It was very cold; there was snow and ice. I saw Professor Süss (the founder of Oberwolfach) and Frau Süss, and also Heinrich Behnke. I remember they asked me to play the piano. It was a beautiful piano - there were two pianos there. The old château at Oberwolfach doesn't exist anymore. I visited Oberwolfach several times after that.Cartan had been invited to the United States in 1942 but he decided that he had to remain in France for the sake of his family, particularly his father who by this time was an old man. He was invited by André Weil to visit Chicago in January 1948, and he received an invitation to visit Harvard University for four months, February to May 1948. He flew into the United States in December 1947 and was met in the airport at New York by Samuel Eilenberg. His major collaboration with Eilenberg was the book Homological Algebra first published in 1956. It is a classic text which has had a profound influence on the subject over a period of nearly half a century.
We explained above that Cartan was appointed professor at the Sorbonne in Paris in November 1940. He taught in Paris from that time until 1969 and then at the Université de Paris-Sud at Orsay from 1970 to 1975. He retired in 1975. At the École Normale Supérieure he started the Séminaire Cartan at the time that Serre was one of his doctoral students. It was Serre who suggested that the seminars should be written up for publication and fifteen ENS-Seminars written by Cartan were published between 1948 and 1964.
Cartan worked on analytic functions, the theory of sheaves, homological theory, algebraic topology and potential theory, producing significant developments in all these areas. Some of his work is put in context by R O Wells Jr reviewing :-
The theory of functions of several complex variables has gone from its infancy with the work of Hartogs, Levi and Poincaré shortly after the turn of the century to its current role as a central field of modern mathematics, much as its predecessor, function theory in one complex variable, did in the 19th century. A central figure in this development has been Henri Cartan, whose series of papers in this field starting in the 1920's dealt with fundamental questions relating to Nevanlinna theory, generalizations of the Mittag-Leffler and Weierstrass theorems to functions of several variables, problems concerned with biholomorphic mappings and the biholomorphic equivalence problem, domains of holomorphy and holomorphic convexity, etc. The major developments in the theory from 1930 to 1950 came from Cartan and his school in France, Behnke's school in Münster, and Oka in Japan. The central ideas up to that time were synthesized in Cartan's Séminaires in the early 1950's, and these were very influential to the next several generations of mathematicians. Cartan's accomplishments were broad and he influenced mathematics through his writing, his teaching, his seminars, and his students in a remarkable manner.In the Preface of  Remmert and Serre write:-
The reader should be aware that these volumes do not fully reflect H Cartan's work, a large part of which is also contained in his fifteen ENS-Seminars (1948-1964) and in his book Homological algebra with S Eilenberg. In particular one cannot appreciate the importance of Cartan's contributions to sheaf theory, Stein manifolds and analytic spaces without studying his 1950/51, 1951/52 and 1953/54 Seminars. Still, we trust that mathematicians throughout the world will welcome the availability of the 'Oeuvres' of a mathematician whose writing and teaching has had such an influence on our generation.We should mention another aspect of Cartan's work which has been involved with politics and in particular supporting human rights. In 1974 a case arose when the Russian authorities placed the mathematician Leonid Plyushch in a special psychiatric hospital. Andrei Sakharov pointed out that this was a political act and Cartan began a strenuous campaign for Plyushch's release. The International Congress of Mathematicians was held in Vancouver in 1974 and this presented an opportunity to gain wide international support for Plyushch with a thousand signatures to a petition for his release. After the Congress Cartan played a major role in setting up the Comité des Mathématiciens to support Plyushch in particular, and all dissident mathematicians. In January 1976 the Soviet authorities released Plyushch which was a major success for Cartan and the Comité des Mathématiciens. But the Committee did not stop after this success. It has supported other mathematicians who have suffered for their political views, such as the Uruguayan mathematician José Luis Massera. For his outstanding work in assisting dissidents Cartan received the Pagels Award from the New York Academy of Sciences.
Cartan is a member of the Académie des Sciences of Paris and of other academies in Europe, the United States, and Japan. He was elected to membership of the Royal Danish Academy of Sciences (1962), The Royal Society of London (1971), the American Academy (1950), the National Academy of Sciences in Washington (1972), The Royal Academy of Belgium (1978), the Akademie der Wissenschaften Göttingen (1971), the Royal Academy of Science Madrid (1971), Bayerische Akademie der Wissenschaften (1974), the Academy of Japan (1979), the Academy of Finland (1979), the Royal Swedish Academy of Sciences (1981), the Polish Academy of Sciences (1985), the Russian Academy (1999). Among the many honours which have been awarded to him was honorary membership of the London Mathematical Society in 1959. He has received honorary doctorates from several universities including ETH Zürich (1955), Münster (1952), Oslo (1961), Sussex (1969), Cambridge (1969), Stockholm (1978), Oxford (1980), Zaragoza (1985), and Athens (1992). He also received the Gold Medal of the National Centre for Scientific research (1976), the Wolf Prize in Mathematics in 1980 and was made Commandeur de la Légion d'Honneur in 1989.
Article by: J J O'Connor and E F Robertson