  The principles of the theory of quantum groups are reviewed from the point of
view of the possibility of their use for deformations of symmetries in physical
models. The R-matrix approach to the theory of quantum groups is discussed in
detail and is taken as the basis of the quantization of classical Lie groups,
as well as some Lie supergroups. We start by laying out the foundations of
non-commutative and non-cocommutative Hopf algebras. Much attention has been
paid to Hecke and Birman-Murakami-Wenzl (BMW) R-matrices and related quantum
matrix algebras. Trigonometric solutions of the Yang-Baxter equation associated
with the quantum groups GL_q(N), SO_q(N), Sp_q(2n) and supergroups GL_q(N|M),
Osp_q(N|2m), as well as their rational (Yangian) limits, are presented.
Rational R-matrices for exceptional Lie algebras and elliptic solutions of the
Yang-Baxter equation are also considered. The basic concepts of the group
algebra of the braid group and its finite dimensional quotients (such as Hecke
and BMW algebras) are outlined. A sketch of the representation theories of the
Hecke and BMW algebras is given (including methods for finding idempotents and
their quantum dimensions). Applications of the theory of quantum groups and
Yang-Baxter equations in various areas of theoretical physics are briefly
discussed.
