Let \(\Omega\) be a bounded symmetric domain in \({\mathbb{C}}^ n\), dv Lebesgue measure, \(L^ 2\) and \(H^ 2\) defined with respect to this measure. For \(f\in L^ 2\), let \(M_ f\) the (unbounded) operator of multiplicaton by f and P the Bergman projection of \(L^ 2\) onto \(H^ 2\). \(H^ 2\) has a reproducing kernel K and associated to it there is a Bergman metric in \(\Omega\). Using this metric one can define ''bounded mean oscillation'' and ''vanishing mean oscillation at the boundary of \(\Omega\) ''. The authors announce the following results: For \(f\in L^ 2\), let \([M_ f,P]\) denote the commutator of \(M_ f\) and P, then a) \([M_ f,P]\) is bounded if and only if f is of bounded mean oscillation in \(\Omega\) ; b) \([M_ f,P]\) is compact if and only if f is of vanishing mean oscillation at the boundary of \(\Omega\).