  The linear stability of MHD Taylor-Couette flows in axially unbounded
cylinders is considered, for magnetic Prandtl number unity. Magnetic fields
varying from purely axial to purely azimuthal are imposed, with a general
helical field parameterized by \beta=B_\phi/B_z. We map out the transition from
the standard MRI for \beta=0 to the nonaxisymmetric Azimuthal MagnetoRotational
Instability (AMRI) for \beta\to \infty. For finite \beta, positive and negative
wave numbers m, corresponding to right and left spirals, are no longer
identical. The transition from \beta=0 to \beta\to\infty includes all the
possible forms of MRI with axisymmetric and nonaxisymmetric modes. For the
nonaxisymmetric modes, the most unstable mode spirals in the opposite direction
to the background field. The standard (\beta=0) MRI is axisymmetric for weak
fields (including the instability with the lowest Reynolds number) but is
nonaxisymmetric for stronger fields. If the azimuthal field is due in part to
an axial current flowing through the fluid itself (and not just along the
central axis), then it is also unstable to the nonaxisymmetric Tayler
instability, which is most effective without rotation. For large \beta this
instability has wavenumber m=1, whereas for \beta\simeq 1 m=2 is most unstable.
The most unstable mode spirals in the same direction as the background field.
