We give sufficient conditions ensuring that any positive classical solution (u,v) of an elliptic system in the whole n-dimensional space has the symmetry property u=v. As an application, we improve some known results on Sobolev-critical elliptic systems of Schrodinger type. Our techniques apply to some supercritical problems as well. We also obtain new Liouville-type theorems for non-cooperative systems. Moreover, we provide several counterexamples which indicate that our assumptions are in a sense necessary. This is a joint work with Philippe Souplet.