A generalization of supersymmetry is proposed based on Z3-graded algebras. Introducing the objects whose ternary commutation relations contain the cubic roots of unity, e2-pi(i)/3, e4-pi(i)/3 and 1, the operators whose trilinear combinations yield the supersymmetric translation generators can be constructed. Cubic matrices forming a ternary algebra are the generalization of Pauli's matrices. The general properties of Z3-grading, some other representations of such algebras, and their possible pertinence with regard to the quark model are briefly discussed.
