  The aim of this paper is to get the boundedness of certain sublinear
operators with rough kernel generated by Calder\'on-Zygmund operators on the
generalized weighted Morrey spaces under generic size conditions which are
satisfied by most of the operators in harmonic analysis. We also prove that the
commutator operators formed by BMO functions and certain sublinear operators
with rough kernel are also bounded on the generalized weighted Morrey spaces.
Marcinkiewicz operator which satisfies the conditions of these theorems can be
considered as an example.
