A numerical semigroup is a submonoid of the non-negative integers, under addition, whose complement in IN is finite. The cardinality of this complement is said to be the genus of the numerical semigroup. In 2008 Bras-Amorós conjectured that the sequence $(n_g)_g$, where $n_g$ is the number of numerical semigroups of genus $g$, behaves like the Fibonacci sequence.
Counting numerical semigroups by genus became then a popular theme and the conjectured behaviour of the sequence $(n_g)_g$ resisted as a conjecture only for 5 years. It became a theorem in a work of Zhai, who also proved other related results.
We intend to survey the vast literature on the theme, mention some of the techniques used, refer some related existing conjectures and mention some new or unexplored approaches.