  We classify all simple strong Harish-Chandra modules for the Lie superalgebra
$W(m,n)$. We show that every such module is either strongly cuspidal or a
module of the highest weight type. We construct tensor modules for $W(m,n)$,
which are parametrized by simple finite-dimensional $gl(m,n)$-modules and show
that every simple strongly cuspidal $W(m,n)$-module is a quotient of a tensor
module. Finally, we realize modules of the highest weight type as simple
quotients of the generalized Verma modules induced from tensor modules for
$W(m-1,n)$.
