His research focused on the areas of real analysis, mathematical physics, the calculus of variations and partial differential equations.

He did not lose his position due to being Jewish, as his previous service as a front-line soldier exempted him; however, his public membership in the social-democratic left was reason enough (for the Nazis) for dismissal.

[4] Courant and David Hilbert authored the influential textbook Methoden der mathematischen Physik, which, with its revised editions, is still current and widely used since its publication in 1924.

[5] Courant's name is also attached to the finite element method,[6] with his numerical treatment of the plain torsion problem for multiply-connected domains, published in 1943.

Courant was an elected member of both the American Philosophical Society (1953) and the United States National Academy of Sciences (1955).

Commenting upon his analysis of experimental results from in-laboratory soap film formations, Courant explained why the existence of a physical solution does not obviate mathematical proof.

[13]In 1912, Courant married Nelly Neumann, who had earned her doctorate at Breslau in synthetic geometry in 1909.