Cooperative interactions within biological macromolecules are of fundamental physiological relevance and have been studied in great detail. Yet, even in the best investigated case of oxygen binding by hemoglobin, our understanding of the structural and thermodynamic bases of cooperativity is far from satisfactory. Several theoretical models have been proposed to explain cooperative O2 binding to hemoglobin, among which the two-state model by Monod, Wyman and Changeux, has been the most successful and the most thoroughly tested. This model explains the functional properties of hemoglobin as resulting from the equilibrium of two quaternary conformations, named R and T, characterized by different ligand affinity, and is capable of very accurate (but not always exact) predictions. This review focuses on the experiments carried out to test the models of cooperativity, and especially the two-state model, and identifies two major deviations, or groups of deviations, between the predictions of this model and the actual experimental results, namely (i) the changes in the behaviour of the T- and R-state due to solvent components; (ii) the appearance of R-like reactivity under experimental conditions in which the T-state should be largely prevalent. Modern models of cooperativity, devised to account for these discrepancies while maintaining the basic two-state hypothesis of Monod, Wyman and Changeux, are also reviewed.