Quantum theory predicts and experiments confirm that nature can produce correlations between distant events that are nonlocal in the sense of violating a Bell inequality 1 . Nevertheless, Bell's strong sentence 'Correlations cry out for explanations' (ref. 2) remains relevant. The maturing of quantum information science and the discovery of the power of non-local correlations, for example for cryptographic key distribution beyond the standard quantum key distribution schemes 3-5 , strengthen Bell's wish and make it even more timely. In 2003, Leggett proposed an alternative model for non-local correlations 6 that he proved to be incompatible with quantum predictions. We present here a new approach to this model, along with new inequalities for testing it. These inequalities can be derived in a very simple way, assuming only the non-negativity of probability distributions; they are also stronger than previously published and experimentally tested Leggett-type inequalities 6-9 . The simplest of the new inequalities is experimentally violated. Then we go beyond Leggett's model, and show that we cannot ascribe even partially defined individual properties to the components of a maximally entangled pair.Formally, a correlation is a conditional probability distribution P(α, β|a, b), where α, β are the outcomes observed by two partners, Alice and Bob, when they make measurements labelled by a and b, respectively. On the abstract level, a and b are merely inputs, freely and independently chosen by Alice and Bob. On a more physical level, Alice and Bob hold two subsystems of a quantum state; in the simple case of qubits, the inputs are naturally characterized by vectors on the Poincaré sphere, hence the notation a,b.How should we understand non-local correlations, in particular those corresponding to entangled quantum states? A natural approach consists in decomposing P(α, β|a, b) into a statistical mixture of hopefully simpler correlations:Bell's locality assumption is P l (α, β|a, b) = P A l (α|a)P B l (β|b), admittedly the simplest choice, but an inadequate one as it turns out: quantum correlations violate Bell's locality 1 . Setting out to explore other choices, it is natural to require first that the P l fulfil the so-called no-signalling condition, that is, that none of the correlations P l results from a communication between Alice and Bob. This can be guaranteed by ensuring spacelike separation between Alice and Bob. Non-signalling correlations happen without any time ordering: there is not a first event, let us say on Alice's side, that causes the second event via some spooky action at a distance. We may phrase it differently: nonsignalling correlations happen from outside space-time, in the sense that there is no story in space-time that tells us how they happen. This is the case in orthodox quantum physics, or in some illuminating toy models such as the non-local box of Popescu and Rohrlich (PR box) 10 . Mathematically, the no-signalling condition reads P l (α|a,b) = P l (α|a) and P l (β|a,b) = P l (β|b): Alice'...