We study the problem of classifying all Poisson-Lie structures on the group Gy of local diffeomorphisms of the real line R¹ which leave the origin fixed, as well as the extended group of diffeomorphisms G₀_∞ ⊃ G_∞ whose action on R¹ does not necessarily fix the origin. A complete classification of all Poisson-Lie structures on the group G_∞ is given. All Poisson-Lie structures of coboundary type on the group G₀_∞ are classified. This includes a classification of all Lie-bialgebra structures on the Lie algebra G_∞ of G_∞, which we prove to be all of coboundary type, and a classification of all Lie-bialgebra structures of coboundary type on the Lie algebra Go_∞ of Go_∞ which is the Witt algebra. A large class of Poisson structures on the space V_λ of λ-densities on the real line is found such that V_λ becomes a homogeneous Poisson space under the action of the Poisson-Lie group G_∞. We construct a series of finite-dimensional quantum groups whose quasiclassical limits are finite-dimensional Poisson-Lie factor groups of G_∞ and G₀_∞. / Ph. D.