Mandelbrot was educated at the École Polytechnique (1945–47) in Paris and at the California Institute of Technology (1947–49). He studied for his doctorate in Paris between 1949 and 1952 and then did research for a year under John von Neumann (von Neumann, John) at the Institute for Advanced Study in Princeton, New Jersey. From 1958 to 1993 he worked for IBM (International Business Machines Corporation) at its Thomas J. Watson Research Center in New York, becoming a research fellow there in 1974. From 1987 he taught at Yale University, becoming the Sterling Professor of Mathematical Sciences in 1999.

As set out in his highly successful book The Fractal Geometry of Nature (1982) and in many articles, Mandelbrot's work is a stimulating mixture of conjecture and observation, both into mathematical processes and their occurrence in nature and in economics. In 1980 he proposed that a certain set governs the behaviour of some iterative processes in mathematics that are easy to define but have remarkably subtle properties. He produced detailed evidence in support of precise conjectures about this set and helped to generate a substantial and continuing interest in the subject. Many of these conjectures have since been proved by others. The set, now called the Mandelbrot set, has the characteristic properties of a fractal: it is very far from being “smooth,” and small regions in the set look like smaller-scale copies of the whole set (a property called self-similarity). Mandelbrot's innovative work with computer graphics stimulated a whole new use of computers in mathematics.

Mandelbrot won a number of awards and honorary degrees. He became a Fellow of the American Academy of Arts and Sciences in 1982 and of the National Academy of Sciences in 1987. He was awarded the Wolf Foundation Prize for Physics in 1993 for his work on fractals, and in 2003 he shared the Japan Prize of the Science and Technology Foundation of Japan for “a substantial contribution to the advance of science and technology.”