In 1923 Schur considered the problem of which polynomials $f\in\mathbb{Z}[X]$ induce bijections on the residue fields $\mathbb{Z}/p\mathbb{Z}$ for infinitely many primes $p$. The authors investigate the analogous question for rational functions, and allow the base field to be any number field.Introduction; Arithmetic-Geometric preparation; Group theoretic exceptionality; Genus 0 condition; Dickson polynomials and Redei functions; Rational functions with Euclidean signature; Sporadic cases of arithmetic exceptionality; Bibliography; Introduction; Arithmetic-Geometric preparation; Group theoretic exceptionality; Genus 0 condition; Dickson polynomials and Redei functions; Rational functions with Euclidean signature; Sporadic cases of arithmetic exceptionality; Bibliography