  We propose a generalization of non-commutative geometry and gauge theories
based on ternary Z_3-graded structures. In the new algebraic structures we
define, we leave all products of two entities free, imposing relations on
ternary products only. These relations reflect the action of the Z_3-group,
which may be either trivial, i.e. abc=bca=cab, generalizing the usual
commutativity, or non-trivial, i.e. abc=jbca, with j=e^{(2\pi i)/3}. The usual
Z_2-graded structures such as Grassmann, Lie and Clifford algebras are
generalized to the Z_3-graded case. Certain suggestions concerning the eventual
use of these new structures in physics of elementary particles are exposed.
