We introduce a Z 3 -graded quantum (2 + 1)-superspace and define Z 3 -graded Hopf algebra structure on algebra of functions on the Z 3 -graded quantum superspace. We construct a differential calculus on the Z 3 -graded quantum superspace, and obtain the corresponding Z 3 -graded Lie superalgebra. We also find a new Z 3 -graded quantum supergroup which is a symmetry group of this calculus.