Ambrosio studied from 1981 Mathematics at the University of Pisa, where he at Ennio De Giorgi summa cum laude his degree ( Laurea ) made in 1985 and until 1988 continued his doctoral studies at the Scuola Normale Superiore. 1988 to 1992 he was a research assistant at the University of Rome ( Tor Vergata). In 1992 he was assistant professor at the University of Pisa and from 1994 professor at the University of Benevento. From 1995 he was a professor at the University of Pavia, before he became professor at the Scuola Normale Superiore in Pisa in 1997. He has been a visiting scientist at the Max Planck Institute for Mathematics in the Sciences in Leipzig, at the Massachusetts Institute of Technology, ETH Zurich and at the IHES.
Ambrosio deals with the calculus of variations, geometric measure theory and the theory of optimal transport with applications to partial differential equations.
In 1991 he received the Prize of the Unione Matematica Bartolozzi Italiana and 1999 whose Caccioppoli Prize. In 1996 he was invited speaker at the 2nd European Congress of Mathematicians in Budapest, and in 2008 he gave a plenary lecture at the European Congress of Mathematicians in Amsterdam (Optimal transportation and evolution problems in spaces of probability measures ). In 2002 he was invited speaker on the ICM in Beijing (Optimal transport maps in Monge - Kantorovich problem). He became a corresponding member of the Accademia dei Lincei and the Istituto Lombardo 2006 2005. In 2003 he was awarded the Fermat Prize.
He was co-editor of Archive for Rational Mechanics and Analysis, Calculus of Variations and Partial Differential Equations of and the Journal of the European Mathematical Society.