For a connected regular scheme X, flat and of finite type over Spec(Z), we construct a reciprocity homomorphism rho_X: C_X -> pi_1^ab(X) which is surjective and whose kernel is the connected component of the identity. The (topological) group C_X is explicitly given and built solely out of data attached to points and curves on X. A similar but weaker statement holds for smooth varieties over finite fields. Our results are based on earlier work of G. Wiesend. This talk is on joint work with M. Kerz.