  A major motivation for the development of semigroup theory was, and still is,
its applications to the study of formal languages. Therefore, it is not
surprising that the correspondence $\mathcal X\mapsto B(\mathcal X)$,
associating to each symbolic dynamical system $\mathcal X$ the formal language
$B(\mathcal X)$ of its blocks, entails a connection between symbolic dynamics
and semigroup theory. In this article we survey some developments on this
connection, since when it was noticed in an article by Almeida, published in
the CIM bulletin, in 2003.
