  We study the problem of classifying all Poisson-Lie structures on the group
$G_{\infty}$ of formal diffeomorphisms of the real line $\zR^{1}$ which leave
the origin fixed, as well as the extended group of diffeomorphisms
$G_{0\infty}\supset G_{\infty}$ whose action on $\zR^{1}$ does not necessarily
fix the origin. A complete local classification of all Poisson-Lie structures
on the groups $G_{\infty}$ and $G_{0\infty}$ is given. This includes a
classification of all Lie-bialgebra structures on the Lie algebra $\Cal
G_{\infty}$ of $G_{\infty}$, which we prove to be all of coboundary type, and a
classification of all Lie-bialgebra strucutures on the Lie algebra $\Cal
G_{0\infty}$ (the Witt algebra) of $G_{0\infty}$ which also turned out to be
all of coboundary type. A large class of Poisson structures on the space
$V_{\lambda}$ of $\lambda$-densities on the real line is found such that
$V_{\lambda}$ becomes a homogeneous Poisson space under the action of the
Poisson-Lie group $G_{\infty}$. We construct a series of quantum semigroups
whose quasiclassical limits are finite-dimensional Poisson-Lie factor groups of
$G_{\infty}$ and $G_{0\infty}$.
