It has been proposed that the observed dark energy can be explained by the effect of large-scale nonlinear inhomogeneities. In this paper, we discuss how observations constrain cosmological models featuring large voids. We start by considering Copernican models, in which the observer is not occupying a special position and homogeneity is preserved on a very large scale. We show how these models, at least in their current realizations, are constrained to give small, but perhaps not negligible in certain contexts, corrections to the cosmological observables. We then examine non-Copernican models, in which the observer is close to the center of a very large void. These models can give large corrections to the observables which mimic an accelerated Friedmann-Lemaitre-Robertson-Walker model. We carefully discuss the main observables and tests able to exclude them.