  We write down a general geometric action principle for spinning strings in
$d$-dimensional Minkowski space, which is formulated without the use of
Grassmann coordinates. Instead, it is constructed in terms of the pull-back of
a left invariant Maurer-Cartan form on the $d$-dimensional Poincar\'e group to
the world sheet. The system contains some interesting special cases. Among them
are the Nambu string (as well as, null and tachyionic strings) where the spin
vanishes, and also the case of a string with a spin current - but no momentum
current. We find the general form for the Virasoro generators, and show that
they are first class constraints in the Hamiltonian formulation of the theory.
The current algebra associated with the momentum and angular momentum densities
are shown, in general, to contain rather complicated anomaly terms which
obstruct quantization. As expected, the anomalies vanish when one specializes
to the case of the Nambu string, and there one simply recovers the algebra
associated with the Poincar\'e loop group. We speculate that there exist other
cases where the anomalies vanish, and that these cases give the bosonization of
the known pseudoclassical formulations of spinning strings.
