  Poisson brackets (P.b) are the natural initial terms for the deformation
quantization of commutative algebras. There is an open problem whether any
Poisson bracket on the polynomial algebra of $n$ variables can be quantized. It
is known (Poincare-Birkhoff-Witt theorem) that any linear P.b. for all $n$ can
be quantized. On the other hand, it is easy to show that in case $n=2$ any P.b.
is quantizable as well. Quadratic P.b. appear as the unitial terms for the
quantization of polynomial algebras as quadratic algebras. The problem of the
quantization of quadratic P.b. is also open. In the paper we show that in case
$n=3$ any quadratic P.b can be quantized. Moreover, the quantization is given
as the quotient algebra of tensor algebra of three variables by relations which
are similar to those in the Poincare-Birkhoff-Witt theorem. The proof uses a
classification of all quadratic Poisson brackets of three variables, which we
also give in the paper. In Appendix we give explicit algebraic constructions of
the quantized algebras appeared here to show that they are related to algebras
of global dimension three considered by M.Artin, W.Schelter, J.Tate, and M.Van
Den Bergh from a different point of view.
