  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
SUq(2) rotator model and its extensions, the construction of deformed exactly
soluble models (u(3)>so(3) model, Interacting Boson Model, Moszkowski model),
the 3-dimensional q-deformed harmonic oscillator and its relation to the
nuclear shell 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 the structure of molecules and of atomic
clusters, as well as an outlook are also given.
