  We construct the quantum versions of the monodromy matrices of KdV theory.
The traces of these quantum monodromy matrices, which will be called as ``${\bf
T}$-operators'', act in highest weight Virasoro modules. The ${\bf
T}$-operators depend on the spectral parameter $\lambda$ and their expansion
around $\lambda = \infty$ generates an infinite set of commuting Hamiltonians
of the quantum KdV system. The ${\bf T}$-operators can be viewed as the
continuous field theory versions of the commuting transfer-matrices of
integrable lattice theory. In particular, we show that for the values
$c=1-3{{(2n+1)^2}\over {2n+3}} , n=1,2,3,... $of the Virasoro central charge
the eigenvalues of the ${\bf T}$-operators satisfy a closed system of
functional equations sufficient for determining the spectrum. For the
ground-state eigenvalue these functional equations are equivalent to those of
massless Thermodynamic Bethe Ansatz for the minimal conformal field theory
${\cal M}_{2,2n+3}$; in general they provide a way to generalize the technique
of Thermodynamic Bethe Ansatz to the excited states. We discuss a
generalization of our approach to the cases of massive field theories obtained
by perturbing these Conformal Field Theories with the operator $\Phi_{1,3}$.
The relation of these ${\bf T}$-operators to the boundary states is also
briefly described.
