He made important contributions in mathematical physics, in particular for the foundations of quantum mechanics and for the theory of partial differential equations.

He studied from 1924 to 1929 at the universities of Graz and Göttingen and received his doctor's degree in 1929 under Richard Courant at Georg August University of Göttingen with the thesis about "Verallgemeinerung der Riemannschen Integrationsmethode auf Differentialgleichungen n-ter Ordnung in zwei Veränderlichen" ("Generalization of Riemann's integration method on differential equations of n-th order in two variables").

When in 1933 the great mathematical-physical tradition in Göttingen terminated with the Machtergreifung of the Nazis, Rellich, having taken an active position against Nazism, was among those forced to leave.

His sister Camilla Juliana Anna was the wife of mathematician Bartel Leendert van der Waerden.

Among Rellich's most important mathematical contributions are his work in the perturbation theory of linear operators on Hilbert spaces: he studied the dependence of the spectral family