  Lie bialgebra structures on $e(2)$ are classified. For two Lie bialgebra
structures which are not coboundaries (i.e. which are not determined by a
classical $r$-matrix) we solve the cocycle condition, find the Lie-Poisson
brackets and obtain quantum group relations. There is one to one correspondence
between Lie bialgebra structures on $e(2)$ and possible quantum deformations of
$U(e(2))$ and $E(2)$.
