The Combinatorial Nullstellensatz is a result published in the late '90 by Alon, that
generalizes to multivariate polynomials the fact that a univariate polynomial of degre
d cannot have d+1 roots. The idea was used before in several papers and known as
the polynomial method. It has found itself very useful to give new proofs, generalizations
ans new results in various fields of mathematics. During this presentation, I will focus on
the various applications in Additive Combinatorics, from the early results: Cauchy-Davenport,
Erdos-Ginzburg-Ziv, Erdos-Heilbronn conjecture, the cyclic case of Snevily's conjecture,up to
some more recent ones.