  Quantum algebras are a mathematical tool which provides us with a class of
symmetries wider than that of Lie algebras, which are contained in the former
as a special case. After a self-contained introduction to the necessary
mathematical tools ($q$-numbers, $q$-analysis, $q$-oscillators, $q$-algebras),
the su$_q$(2) rotator model and its extensions, the construction of deformed
exactly soluble models (Interacting Boson Model, Moszkowski model), the use of
deformed bosons in the description of pairing correlations, and the symmetries
of the anisotropic quantum harmonic oscillator with rational ratios of
frequencies, which underly the structure of superdeformed and hyperdeformed
nuclei, are discussed in some detail. A brief description of similar
applications to molecular structure and an outlook are also given.
