Quantum theory is successfully tested in any experimental lab every day. Apart from its experimental validity, quantum theory also constitutes a robust theoretical framework: small variations of its formalism often lead to highly implausible consequences, such as violation of the no-signalling principle or a significant increase of the computational power. In fact, it has been argued that quantum theory may represent an island in theory space. We show that, at the level of correlations, quantum theory may not be as special as initially thought. In order to do so, we define the set of almost quantum correlations and prove that this set is (i) strictly larger than the set of quantum correlations but (ii) satisfies most of the information principles introduced to characterize quantum correlations, such as local orthogonality, macroscopic locality, no advantage for nonlocal computation or non-trivial communication complexity. We also provide numerical evidence that the set is compatible with information causality. Finally, we briefly discuss how the set of almost quantum correlations naturally emerges in the consistent histories approach to quantum physics.