  For the algebra $I_n$ of polynomial integro-differential operators over a
field $K$ of characteristic zero, a classification of simple weight and
generalized weight (left and right) $I_n$-modules is given. It is proven that
the category of weight $I_n$-modules is semisimple. An explicit description of
generalized weight $I_n$-modules is given and using it a criterion is obtained
for the problem of classification of indecomposable generalized weight
$I_n$-modules to be of finite representation type, tame or wild. In the tame
case, a classification of indecomposable generalized weight $I_n$-modules is
given. In the wild case `natural` tame subcategories are considered with
explicit description of indecomposable modules. It is proven that every
generalized weight $I_n$-module is a unique sum of absolutely prime modules.
For an arbitrary ring $R$, we introduce the concept of {\em absolutely prime}
$R$-module (a nonzero $R$-module $M$ is absolutely prime if all nonzero
subfactors of $M$ have the same annihilator). It is shown that every
indecomposable generalized weight $I_n$-module is equidimensional. A criterion
is given for a generalized weight $I_n$-module to be finitely generated.
