  Some introductory concepts and basic definitions of the Lie superalgebras and
their quantum deformations are exposed. Especially the induced representation
methods in both cases are described. Based on the Kac representation theory we
have succeeded in constructing representations of several higher rank
superalgebras. When representations of quantum superalgebras are concerned, we
have developed a method which can be applied not only to the one-parametric
quantum deformations but also to the multi-parametric ones. As an intermediate
step the Gel'fand-Zetlin basis description is extended to the case of
superalgebras and their quantum deformations. Our approach also allows us to
establish in consistent ways defining relations of quantum (super)algebras.
Some illustrations are given.
