  In this paper we introduce generalized grand Morrey spaces in the framework
of quasimetric measure spaces, in the spirit of the so-called grand Lebesgue
spaces. We prove a kind of reduction lemma which is applicable to a variety of
operators to reduce their boundedness in generalized grand Morrey spaces to the
corresponding boundedness in Morrey spaces, as a result of this application, we
obtain the boundedness of the Hardy-Littlewood maximal operator as well as the
boundedness of Calder\'on-Zygmund potential type operators. Boundedness of
Riesz type potential operators are also obtained in the framework of
homogeneous and also in the nonhomogeneous case in generalized grand Morrey
spaces.
