  We discuss proofs of nonlocality based on a generalization by Erwin
Schr\"odinger of the argument of Einstein, Podolsky and Rosen. These proofs do
not appeal in any way to Bell's inequalities. Indeed, one striking feature of
the proofs is that they can be used to establish nonlocality solely on the
basis of suitably robust perfect correlations. First we explain that
Schr\"odinger's argument shows that locality and the perfect correlations
between measurements of observables on spatially separated systems implies the
existence of a non-contextual value-map for quantum observables; non-contextual
means that the observable has a particular value before its measurement, for
any given quantum system, and that any experiment "measuring this observable"
will reveal that value. Then, we establish the impossibility of a
non-contextual value-map for quantum observables {\it without invoking any
further quantum predictions}. Combining this with Schr\"odinger's argument
implies nonlocality. Finally, we illustrate how Bohmian mechanics is compatible
with the impossibility of a non-contextual value-map.
