From 1904 to 1908 he studied in Göttingen and Munich, mainly mathematics and physics. His doctorate was presented to him at Göttingen under the direction of Hilbert and Minkowski. In 1910, he obtained a teaching post of private lecturer at Göttingen.
From 1913 to 1930 he held the chair of mathematics at the Technische Hochschule of Zurich. In 1928 and 1929 he was a visiting professor at Princeton University and in 1930 he became Hilbert's successor at Göttingen where he held the chair of mathematics. With the rise of the National Socialist in 1933, he left Germany for the U.S. where he worked with Einstein at the Institute for Advanced Study at Princeton University until his retirement in 1952.
He tried to incorporate electromagnetism in the geometrical formalism of general relativity. In 1913, Weyl published Die Idee der Riemannschen Fläche which unified analysis, geometry and topology. He produced the first gauge theory in which the electromagnetic field and the gravitational field appear as geometrical properties of spacetime.
From 1923 to 1938 he developed the concept of continuous groups in terms of matrix representations. He established a group-theoretic basis for quantum mechanics. He also showed how to use exponential sums in diophantine approximation, with his criterion for uniform distribution mode 1, which was fundamental step in analytic number theory.
Weyl's own comment, although half a joke, sums up his personality.