  We describe the set of generalized Poincare and conformal superalgebras in
D=4,5 and 7 dimensions as two sequences of superalgebraic structures, taking
values in the division algebras R, C and H. The generalized conformal
superalgebras are described for D=4 by OSp(1;8| R), for D=5 by SU(4,4;1) and
for D=7 by U_\alpha U(8;1|H). The relation with other schemes, in particular
the framework of conformal spin (super)algebras and Jordan (super)algebras is
discussed. By extending the division-algebra-valued superalgebras to octonions
we get in D=11 an octonionic generalized Poincare superalgebra, which we call
octonionic M-algebra, describing the octonionic M-theory. It contains 32 real
supercharges but, due to the octonionic structure, only 52 real bosonic
generators remain independent in place of the 528 bosonic charges of standard
M-algebra. In octonionic M-theory there is a sort of equivalence between the
octonionic M2 (supermembrane) and the octonionic M5 (super-5-brane) sectors. We
also define the octonionic generalized conformal M-superalgebra, with 239
bosonic generators.
