  A ribbon is a surface swept out by a line segment turning as it moves along a
central curve. For narrow magnetic ribbons, for which the length of the line
segment is much less than the length of the curve, the anisotropy induced by
the magnetostatic interaction is biaxial, with hard axis normal to the ribbon
and easy axis along the central curve. The micromagnetic energy of a narrow
ribbon reduces to that of a one-dimensional ferromagnetic wire, but with
curvature, torsion and local anisotropy modified by the rate of turning. These
general results are applied to two examples, namely a helicoid ribbon, for
which the central curve is a straight line, and a M\"obius ribbon, for which
the central curve is a circle about which the line segment executes a
$180^\circ$ twist. In both examples, for large positive tangential anisotropy,
the ground state magnetization lies tangent to the central curve. As the
tangential anisotropy is decreased, the ground state magnetization undergoes a
transition, acquiring an in-surface component perpendicular to the central
curve. For the helicoid ribbon, the transition occurs at vanishing anisotropy,
below which the ground state is uniformly perpendicular to the central curve.
The transition for the M\"obius ribbon is more subtle; it occurs at a positive
critical value of the anisotropy, below which the ground state is nonuniform.
For the helicoid ribbon, the dispersion law for spin wave excitations about the
tangential state is found to exhibit an asymmetry determined by the geometric
and magnetic chiralities.
