The thesis investigates the locality axiom of general local covariance for Abelian gauge theories. Two models, Maxwell k-forms (higher analogues of the electromagnetic vector potential) and the U(1) Yang-Mills model are analyzed over globally hyperbolic spacetimes. Our attention is mainly focused on the locality axiom of general local covariance, which states that a causal embedding between spacetimes should induce an inclusion at the level of observables. Both at the classical and at the quantum level, it turns out that the models we consider violate locality depending on certain global features of the background spacetime. For Maxwell k-forms, we prove that there is no coherent way to recover the locality axiom. For the U(1) Yang-Mills model we adopt two different approaches: in the first one locality can be recovered coherently, but the class of observables we consider fails in detecting those field configurations which correspond to the Aharonov-Bohm effect; conversely, in our second approach observables are defined in the spirit of Wilson loops (hence capturing also Aharonov-Bohm configurations), but a no-go theorem shows that locality cannot be recovered in a coherent way.