  We use the Hamiltonian framework to study massless QCD$_{1+1}$, i.e.\
Yang-Mills gauge theories with massless Dirac fermions on a cylinder (= (1+1)
dimensional spacetime $S^1\times \R$) and make explicite the full,
non-perturbative structure of these quantum field theory models. We consider
$N_F$ fermion flavors and gauge group either $\U(N_C)$, $\SU(N_C)$ or another
Lie subgroup of $\U(N_C)$. In this approach, anomalies are traced back to
kinematical requirements such as positivity of the Hamiltonian, gauge
invariance, and the condition that all observables are represented by
well-defined operators on a Hilbert space. We also give equal time commutators
of the energy momentum tensor and find a gauge-covariant form of the (affine-)
Sugawara construction. This allows us to represent massless QCD$_{1+1}$ as a
gauge theory of Kac-Moody currents and prove its equivalence to a gauged
Wess-Zumino-Witten model with a dynamical Yang-Mills field.
