  In this thesis some systems with exotic symmetries, which are a peculiarity
in 2+1 space-time dimensions, are studied. Coded in the exotic structure, there
appear noncommutative coordinates and a phases structure. This kind of systems
has attracted attention from different areas of physics independently. Among
them, we can mention: theory of ray representations of Lie groups, anyon
physics, some condensed matter systems, for instance the quantum Hall effect,
planar gauge and gravitation theories, noncommutative field theory,
noncommutative geometry, and noncommutative quantum mechanics. We focus our
study in some topics on exotic nonrelativistic symmetries, such as the exotic
Newton-Hooke group, the relation between such systems and the noncommutative
Landau problem, the symmetries of noncommutative Landau problem, and of its
supersymmetric extension. The exotic Newton-Hooke group corresponds to the
nonrelativistic limit of the de Sitter groups, and has as a particular, flat
limit the exotic Galilei group. For the exotic Newton-Hooke symmetry, we
construct a free particle action, and realize a complete study of its classical
and quantum properties. The Newton-Hooke system is intimately related with the
noncommutative Landau problem, which we study apart. We show that the inclusion
of spin degrees of freedom in the noncommutative Landau problem produces a
natural integration of the exotic Newton-Hooke group with the conformal
symmetry and supersymmetry.
