  We introduce a new family of affine $W$-algebras associated with the
centralizers of arbitrary nilpotent elements in $\mathfrak{gl}_N$. We define
them by using a version of the BRST complex of the quantum Drinfeld--Sokolov
reduction. A family of free generators of the new algebras is produced in an
explicit form. We also give an analogue of the Fateev--Lukyanov realization for
these algebras by applying a Miura-type map.
