Trèves earned his Ph.D. in 1958 from Paris-Sorbonne University under the supervision of Laurent Schwartz. He then went to the United States where from 1958 to 1960 he was assistant professor at the University of California, Berkeley. From 1961 to 1964 he was an associate professor at Yeshiva University, and from 1964 to 1970 professor at Purdue University. In 1970 he became a professor at Rutgers University, and then, in 1984, Robert-Adrian professor of mathematics. He became professor emeritus in 2005.
In 1972 he received the Chauvenet Prize for "On local solvability of linear partial differential equations" in the Bulletin of the AMS (Volume 76, 1970, pp. 552–571). It was about the problem he worked in 1962 with Louis Nirenberg with whom he found necessary and sufficient conditions for the solvability of equations with analytic coefficients, 1969 (Comptes Rendus de l'Académie des Sciences Paris Bd.269). The question was first presented to him in 1955 by Schwartz as a thesis problem.
In 1977 he was Guggenheim Fellow. In 1991 he received the Leroy P. Steele Prize for his book on pseudo-differential operators and Fourier integral operators. In 2003 he became a foreign member of the Brazilian Academy of Sciences. In 1970 he was an invited speaker at the International Congress of Mathematicians in Nice (Hamiltonian fields, bicharacteristic strips in relation with existence and regularity of solutions of linear partial differential equations). He is a fellow of the American Mathematical Society.
- "On the theory of linear partial differential operators with analytic coefficients." Transactions of the American Mathematical Society 137 (1969): 1–20. doi:10.2307/1994784
- "An abstract nonlinear Cauchy-Kovalevska theorem." Transactions of the American Mathematical Society 150, no. 1 (1970): 77–92. MR0274911
- "Differential polynomials and decay at infinity." Bulletin of the American Mathematical Society 66, no. 3 (1960): 184–186. MR0117448
- "Discrete phenomena in uniqueness in the Cauchy problem." Proceedings of the American Mathematical Society 46, no. 2 (1974): 229–233. MR0352679
- with Howard Jacobowitz: "Nowhere solvable homogeneous partial differential equations." Bulletin of the American Mathematical Society 8, no. 3 (1983): 467–469. MR693964
- with Nicholas Hanges: "On the analyticity of solutions of first-order nonlinear PDE." Transactions of the American Mathematical Society 331, no. 2 (1992): 627–638. MR1061776
- Locally Convex Spaces and Linear Partial Differential Equations.. Grundlehren der Mathematischen Wissenschaften, Springer-Verlag 1967.
- Linear Partial Differential Equations with Constant Coefficients. Harwood Academic 1968.
- Introduction to Pseudodifferential and Fourier Integral Operators. two volumrd, Kluwer/Plenum, Springer-Verlag 1980, 1981.
- Topological Vector Spaces, Distributions and Kernels. Dover 2006, ISBN 0-486-45352-9.
- Basic Linear Partial Differential Equations. Academic Press 1975, Dover 2006.
- With Paulo Cordaro: Hyperfunctions on Hypo-analytic Manifolds. Princeton, Annals of Mathematical Studies 1994.
- Hypo-Analytic Structures – Local Theory. Princeton University Press 1992.
- Homotopy Formulas in the Tangential Cauchy-Riemann Complex. Memoirs AMS 1990.
- Trèves, F. "Hamiltonian fields, bicharactertistic strips in relation with existence and regularity of solutions of linear partial differential equations." In Actes, Congrès Intern. Math, vol. 2, pp. 803–811. 1970.
- List of Fellows of the American Mathematical Society, retrieved 2013-11-26.
- Beals, Richard (1977). "Review: Basic linear partial differential equations by François Trèves" (PDF). Bull. Amer. Math. Soc. 83 (2): 208–211. doi:10.1090/s0002-9904-1977-14265-1.
- Schapira, Pierre (1996). "Review: Hyperfunctions on hypo-analytic manifolds by Paulo D. Cordaro and François Trèves" (PDF). Bull. Amer. Math. Soc. (N.S.). 33 (1): 115–118. doi:10.1090/s0273-0979-96-00628-3.
- Webster, Sidney M. (1994). "Review: Hypo-analytic structures, local theory by François Trèves" (PDF). Bull. Amer. Math. Soc. (N.S.). 30 (2): 290–292. doi:10.1090/s0273-0979-1994-00468-9.