  We consider 2-dimensional QCD on a cylinder, where space is a circle of
length $L$. We formulate the theory in terms of gauge-invariant gluon operators
and multiple-winding meson (open string) operators. The meson bilocal operators
satisfy a $W_\infty$ current algebra. The gluon sector (closed strings)
contains purely quantum mechanical degrees of freedom. The description of this
sector in terms of non-relativistic fermions leads to a $W_\infty$ algebra. The
spectrum of excitations of the full theory is therefore organized according to
two different algebras: a wedge subalgebra of $W_\infty$ current algebra in the
meson sector and a wedge subalgebra of $W_\infty$ algebra in the glueball
sector. In the large $N$ limit the theory becomes semiclassical and an
effective description for the gluon degrees of freedom can be obtained. We have
solved the effective theory of the gluons in the small $L$ limit. We get a
glueball spectrum which coincides with the `discrete states' of the (Euclidean)
$c=1$ string theory. We remark on the implications of these results for (a) QCD
at finite temperature and (b) string theory.
