  The aim of this paper is to give an overview and to compare the different
deformation theories of algebraic structures. We describe in each case the
corresponding notions of degeneration and rigidity. We illustrate these notions
with examples and give some general properties. The last part of this work
shows how these notions help in the study of associative algebras varieties.
The first and popular deformation approach was introduced by M. Gerstenhaber
for rings and algebras using formal power series. A noncommutative version was
given by Pinczon and generalized by F. Nadaud. A more general approach called
global deformation follows from a general theory of Schlessinger and was
developed by A. Fialowski in order to deform infinite dimensional nilpotent Lie
algebras. In a Nonstandard framework, M.Goze introduced the notion of
perturbation for studying the rigidity of finite dimensional complex Lie
algebras. All these approaches has in common the fact that we make an
"extension" of the field. These theories may be applied to any multilinear
structure. We shall be concerned in this paper with the category of associative
algebras.
