  The concepts of symmetry and symmetry groups are at the heart of several
developments in modern theoretical and mathematical physics. The present paper
is devoted to a number of selected topics within this framework: Euclidean and
rotation groups; the properties of fullerenes in physical chemistry; Galilei,
Lorentz and Poincare groups; conformal transformations and the Laplace
equation; quantum groups and Sklyanin algebras. For example, graphite can be
vaporized by laser irradiation, producing a remarkably stable cluster
consisting of 60 carbon atoms. The corresponding theoretical model considers a
truncated icosahedron, i.e. a polygon with 60 vertices and 32 faces, 12 of
which are pentagonal and 20 hexagonal. The Carbon 60 molecule obtained when a
carbon atom is placed at each vertex of this structure has all valences
satisfied by two single bonds and one double bond. In other words, a structure
in which a pentagon is completely surrounded by hexagons is stable. Thus, a
cage in which all 12 pentagons are completely surrounded by hexagons has
optimum stability. On a more formal side, the exactly solvable models of
quantum and statistical physics can be studied with the help of the quantum
inverse problem method. The problem of enumerating the discrete quantum systems
which can be solved by the quantum inverse problem method reduces to the
problem of enumerating the operator-valued functions that satisfy an equation
involving a fixed solution of the quantum Yang--Baxter equation. Two basic
equations exist which provide a systematic procedure for obtaining completely
integrable lattice approximations to various continuous completely integrable
systems. This analysis leads in turn to the discovery of Sklyanin algebras.
