This paper surveys the area of Free Burnside Semigroups. The theory of
these semigroups, as is the case for groups, is far from being completely
known. For semigroups, the most impressive results were obtained in the
last 10 years. In this paper we give priority to the mathematical treatment of the problem
and do not stress too much neither motivation nor the historical aspects.
No proofs are presented in this paper, but we tried to give as many examples
as was possible.