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This paper is a survey on the study of the behaviour of the composition ofpolynomials on the computation of Gr\"obner bases. This survey brings togethersome works published between 1995 and 2007. The authors of these papers gaveanswers to some questions in this subject for several types of Gr\"obner bases,over different monomials orderings and over different polynomials rings. Someof these answers are complete and some are not. Some papers of them were givento answer some open questions given in the first published paper of thesepapers and other some were given to generalize previous work. Note that, thereare many other works in this subject over other special cases and types ofGr\"obner bases under the usual definition over arbitrary fields, over othertypes of bases, under the non-commutative case and with other methods of proofsand calculations.

Gr\"obner Bases under Composition, Survey paper