  This is a brief review of our recent work attempted at a generalization of
the Grassmann algebra to the paragrassmann ones. The main aim is constructing
an algebraic basis for representing `fractional' symmetries appearing in $2D$
integrable models and also introduced earlier as a natural generalization of
supersymmetries. We have shown that these algebras are naturally related to
quantum groups with $q = {\rm root \;of \; unity}$. By now we have a general
construction of the paragrassmann calculus with one variable and preliminary
results on deriving a natural generalization of the Neveu--Schwarz--Ramond
algebra. The main emphasis of this report is on a new general construction of
paragrassmann algebras with any number of variables, N. It is shown that for
the nilpotency indices $(p + 1) = 3, 4, 6$ the algebras are almost as simple as
the Grassmann algebra (for which $(p + 1) = 2$). A general algorithm for
deriving algebras with arbitrary p and N is also given. However, it is shown
that this algorithm does not exhaust all possible algebras, and the simplest
example of an `exceptional' algebra is presented for $p = 4, N = 4$.
