  We develop a general theory of $W$-algebras in the context of supersymmetric
vertex algebras. We describe the structure of $W$-algebras associated with odd
nilpotent elements of Lie superalgebras in terms of their free generating sets.
As an application, we produce explicit free generators of the $W$-algebra
associated with the odd principal nilpotent element of the Lie superalgebra
$\mathfrak{gl}(n+1|n).$
