  In the paper "On some unsolved problems in quantum group theory", V.Drinfeld
formulated the problem of the existence of a universal quantization for Lie
bialgebras. When the paper "Tensor structures arising from affine Lie algebras,
III", by Kazhdan and Lusztig, appeared, Drinfeld asked whether its methods
could be useful for the problem of universal quantization of Lie bialgebras. In
this paper we use these methods to construct the universal quantization, which
gives a positive answer to Drinfeld's question. We also show the existence of
universal quantization of classical r-matrices, unitary r-matrices, and
quasitriangular Lie bialgebras, which answers the corresponding questions of
Drinfeld.
