  We study weight modules of the Lie algebra $W_2$ of vector fields on
${\mathbb C}^2$. A classification of all simple weight modules of $W_2$ with a
uniformly bounded set of weight multiplicities is provided. To achieve this
classification we introduce a new family of generalized tensor $W_n$-modules.
Our classification result is an important step in the classification of all
simple weight $W_n$-modules with finite weight multiplicities.
