  After a short introduction on Clifford algebras of polynomials, we give a
general method of constructing a matrix representation. This process of
linearization leads naturally to two fundamental structures: the generalized
Clifford algebra (GCA) and the generalized Grassmann algebra (GGA) which are
studied. Then, it is proved that if we equip the GGA with a differential
structure, we obtain the $q-$deformed Heisenberg algebra or the
$q-$oscillators. Finally, it is shown that the $q-$deformed Heisenberg algebra
is the basic tool to define an adapted superspace leading to the extension of
supersymmetry called fractional supersymmetry of order $F$ (FSUSY), $F=2$
corresponding to the usual supersymmetry. Local FSUSY in one dimension is then
contructed in the world-line formalism, and an extension of the Dirac equation
is obtained. In two dimensions, it turns out that FSUSY is a conformal field
theory and in addition to the stress energy tensor, a supercurrent of conformal
weight $1+1/F$, which generates a symmetry between the primary fields of
conformal weight $(0,1, \cdots, 1-1/F$), is obtained. The algebra is explicitly
constructed. We also show that in $1+2$ dimensions FSUSY is a non-trivial
extension of the Poincar\'e algebra which generates a symmetry among fractional
spin states or anyons. Unitarity of the representation is checked. Finally, we
prove that, independently of the dimension, a natural classification emerges
according to the decomposition of $F$ as a product of prime numbers and that
FSUSY is a symmetry which closes non-linearly, and is sustained by mathematical
structures that go beyond Lie or super-Lie algebras.
