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For the algebra $I_n$ of polynomial integro-differential operators over afield $K$ of characteristic zero, a classification of simple weight andgeneralized weight (left and right) $I_n$-modules is given. It is proven thatthe category of weight $I_n$-modules is semisimple. An explicit description ofgeneralized weight $I_n$-modules is given and using it a criterion is obtainedfor the problem of classification of indecomposable generalized weight$I_n$-modules to be of finite representation type, tame or wild. In the tamecase, a classification of indecomposable generalized weight $I_n$-modules isgiven. In the wild case `natural` tame subcategories are considered withexplicit description of indecomposable modules. It is proven that everygeneralized 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 nonzerosubfactors of $M$ have the same annihilator). It is shown that everyindecomposable generalized weight $I_n$-module is equidimensional. A criterionis given for a generalized weight $I_n$-module to be finitely generated.

Explicit description of generalized weight modules of the algebra of polynomial integro-differential operators $I_n$