  We study existence and properties of ground states for the nonlinear
Schr\"odinger equation with combined power nonlinearities \[ -\Delta u= \lambda
u + \mu |u|^{q-2} u + |u|^{2^*-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 3$,}
\] having prescribed mass \[ \int_{\mathbb{R}^N} |u|^2 = a^2, \] in the
\emph{Sobolev critical case}. For a $L^2$-subcritical, $L^2$-critical, of
$L^2$-supercritical perturbation $\mu |u|^{q-2} u$ we prove several
existence/non-existence and stability/instability results.
  This study can be considered as a counterpart of the Brezis-Nirenberg problem
in the context of normalized solutions, and seems to be the first contribution
regarding existence of normalized ground states for the Sobolev critical NLSE
in the whole space $\mathbb{R}^N$.
