  In this paper we classify all simple weight modules for a quantum group $U_q$
at a complex root of unity $q$ when the Lie algebra is not of type $G_2$. By a
weight module we mean a finitely generated $U_q$-module which has finite
dimensional weight spaces and is a sum of those. Our approach follows the
procedures used by S. Fernando and O. Mathieu to solve the corresponding
problem for semisimple complex Lie algebras.
