  We numerically investigate the efficiency of a spherical Couette flow at
generating a self-sustained magnetic field. No dynamo action occurs for
axisymmetric flow while we always found a dynamo when non-axisymmetric
hydrodynamical instabilities are excited. Without rotation of the outer sphere,
typical critical magnetic Reynolds numbers $Rm_c$ are of the order of a few
thousands. They increase as the mechanical forcing imposed by the inner core on
the flow increases (Reynolds number $Re$). Namely, no dynamo is found if the
magnetic Prandtl number $Pm=Rm/Re$ is less than a critical value $Pm_c\sim 1$.
Oscillating quadrupolar dynamos are present in the vicinity of the dynamo
onset. Saturated magnetic fields obtained in supercritical regimes (either
$Re>2 Re_c$ or $Pm>2Pm_c$) correspond to the equipartition between magnetic and
kinetic energies. A global rotation of the system (Ekman numbers $E=10^{-3},
10^{-4}$) yields to a slight decrease (factor 2) of the critical magnetic
Prandtl number, but we find a peculiar regime where dynamo action may be
obtained for relatively low magnetic Reynolds numbers ($Rm_c\sim 300$). In this
dynamical regime (Rossby number $Ro\sim -1$, spheres in opposite direction) at
a moderate Ekman number ($E=10^{-3}$), a enhanced shear layer around the inner
core might explain the decrease of the dynamo threshold. For lower $E$
($E=10^{-4}$) this internal shear layer becomes unstable, leading to small
scales fluctuations, and the favorable dynamo regime is lost. We also model the
effect of ferromagnetic boundary conditions. Their presence have only a small
impact on the dynamo onset but clearly enhance the saturated magnetic field in
the ferromagnetic parts. Implications for experimental studies are discussed.
