  Decades ago S. Lundquist, S. Chandrasekhar, P.H. Roberts and R. J.~Tayler
first posed questions about the stability of Taylor-Couette flows of conducting
material under the influence of large-scale magnetic fields. These and many new
questions can now be answered numerically where the nonlinear simulations even
provide the instability-induced values of several transport coefficients. The
cylindrical containers are axially unbounded and penetrated by magnetic
background fields with axial and/or azimuthal components. The influence of the
magnetic Prandtl number $Pm$ on the onset of the instabilities is shown to be
substantial. The potential flow subject to axial fields becomes unstable
against axisymmetric perturbations for a certain supercritical value of the
averaged Reynolds number $\overline{Rm}=\sqrt{Re\cdot Rm}$ (with $Re$ the
Reynolds number of rotation, $Rm$ its magnetic Reynolds number). Rotation
profiles as flat as the quasi-Keplerian rotation law scale similarly but only
for $Pm\gg 1$ while for $Pm\ll 1$ the instability instead sets in for
supercritical $Rm$ at an optimal value of the magnetic field. Among the
considered instabilities of azimuthal fields, those of the Chandrasekhar-type,
where the background field and the background flow have identical radial
profiles, are particularly interesting. They are unstable against
nonaxisymmetric perturbations if at least one of the diffusivities is non-zero.
For $Pm\ll 1$ the onset of the instability scales with $Re$ while it scales
with $\overline{Rm}$ for $Pm\gg 1$. - Even superrotation can be destabilized by
azimuthal and current-free magnetic fields; this recently discovered
nonaxisymmetric instability is of a double-diffusive character, thus excluding
$Pm= 1$. It scales with $Re$ for $Pm\to 0$ and with $Rm$ for $Pm\to \infty$.
