  A finite group $G$ is called a Schur group, if any Schur ring over $G$ is
associated in a natural way with a subgroup of $Sym(G)$ that contains all right
translations. Recently, the authors have completely identified the cyclic Schur
groups. In this paper it is shown that any abelian Schur group belongs to one
of several explicitly given families only. In particular, any non-cyclic
abelian Schur group of odd order is isomorphic to $Z_3\times Z_{3^k}$ or
$Z_3\times Z_3\times Z_p$ where $k\ge 1$ and $p$ is a prime. In addition, we
prove that $Z_2\times Z_2\times Z_p$ is a Schur group for every prime $p$.
