The original goal of the EPR paper was to show that quantum mechanics is incomplete. Hence, that extra variables have to be added to complete it, contrary to what Cedric claims. The goal of EPR is to show that either nature is non-local (and thus in conflict with SR) either quantum mechanics is incomplete. Since Einstein was not ready to abandon locality and SR, he concluded that quantum mechanics is incomplete.
Later however, John Bell would show that quantum mechanics is in fact non-local. To do that, he first devised an inequality that would have to be satisfied by any local physical theory. Then, he showed that this inequality is violated for certain entangled states, thereby proving that quantum mechanics is non-local. In the 70's Alain Aspect then made an experiment to check if Bell's inequalities were violated in nature or not. There have been many similar experiments since then and they all point to nature being non-local and quantum mechanics being a good description of this non-locality.
Now, there are possible loopholes in the experiments, which I won't discuss here.
One can also object that quantum mechanics described by Schrödinger's equation is not Lorentz-invariant, so we should not expect quantum mechanics to agree with SR.
What about equations which are Lorentz-invariant? Dirac equations, Klein-Gordon, etc... That's where it gets difficult. We know that a correct description of these equations requires quantum field theories. But we only manage to treat the field theories perturbatively. Other approaches are numerical or very limited. So I don't know of any detailed study within the context of quantum field theory of entanglement and Bell inequalities. But I hope someone can come in with more information about these. My knowledge is limited in these areas.