Let S be an operator system -- a self-adjoint linear subspace of a unital C*-algebra A 
such that contains 1 and A=C*(S) is generated by S.  A boundary representation for S 
is an irreducible representation \pi of C*(S) on a Hilbert space 
with the property that $\pi\restriction_S$ has a unique completely positive extension 
to C*(S).  The set $\partial_S$ of all (unitary equivalence classes of) boundary 
representations is the noncommutative counterpart of the Choquet boundary of a 
function system $S\subseteq C(X)$ that separates points of X.  


It is known that the closure of the Choquet boundary of a function system S is 
the Silov boundary of X relative to S.  The corresponding noncommutative problem 
of whether every operator system has  ``sufficiently many" boundary representations 
was formulated in 1969, but has remained unsolved despite progress on related issues.  
In particular, it was unknown if $\partial_S$ is nonempty for generic S.  In this 
paper we show that every separable  operator system has sufficiently many boundary 
representations.  Our methods use separability in an essential way.