  We initiate and develop the theory of finite $W$-superalgebras
$\mathcal{W}_\chi$ associated to the queer Lie superalgebra $\g=\q(N)$ and a
nilpotent linear functional $\chi \in \ev\g^*$. We show that the definition of
the $W$-superalgebra is independent of various choices. We also establish a
Skryabin type equivalence between the category of $\mathcal{W}_\chi$-modules
and a category of certain $\g$-modules.
