The Duffin-Schaeffer conjecture in Diophantine approximation [R. J. Duffin and A. C. Schaeffer, Duke Math. J. 8, 243-255 (1941; Zbl 0025.11002)] has been a challenging open problem for almost half a century. The present paper provides a proof for a k-dimensional version (k\(\geq 2\) due to V. G. Sprindzhuk) as well as a further contribution to the case \(k=1\). The proof makes use of several results from previous attempts to master the conjecture, e.g. a generalization of Gallagher’s ergodic theorem due to V. T. Vil’chinskij (which reduces the problem to the case of measure 0 or 1).