  We define and study the ternary analogues of Clifford algebras. It is proved
that the ternary Clifford algebra with $N$ generators is isomorphic to the
subalgebra of the elements of grade zero of the ternary Clifford algebra with
$N+1$ generators. In the case $N=3$ the ternary commutator of cubic matrices
induced by the ternary commutator of the elements of grade zero is derived. We
apply the ternary Clifford algebra with one generator to construct the
$Z_3$-graded generalization of the simplest algebra of supersymmetries.
