  We study existence and properties of ground states for the nonlinear
Schr\"odinger equation with combined power nonlinearities \[ -\Delta u= \lambda
u + \mu |u|^{q-2} u + |u|^{p-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 1$,}
\] having prescribed mass \[ \int_{\mathbb{R}^N} |u|^2 = a^2. \] Under
different assumptions on $q<p$, $a>0$ and $\mu \in \mathbb{R}$ we prove several
existence and stability/instability results. In particular, we consider cases
when \[ 2<q \le 2+ \frac{4}{N} \le p<2^*, \quad q \neq p, \] i.e. the two
nonlinearities have different character with respect to the $L^2$-critical
exponent. These cases present substantial differences with respect to purely
subcritical or supercritical situations, which were already studied in the
literature.
  We also give new criteria for global existence and finite time blow-up in the
associated dispersive equation.
