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We classify all simple $W_n$-modules with finite-dimensional weight spaces.Every such module is either of a highest weight type or is a quotient of amodule of tensor fields on a torus, which was conjectured by Eswara Rao. Thisgeneralizes the classical result of Mathieu on simple weight modules for theVirasoro algebra. In our proof of the classification we construct a functorfrom the category of cuspidal $W_n$-modules to the category of $W_n$-moduleswith a compatible action of the algebra of functions on a torus. We alsopresent a new identity for certain quadratic elements in the universalenveloping algebra of $W_1$, which provides important information aboutcuspidal $W_1$-modules.

Classification of simple $W_n$-modules with finite-dimensional weight spaces