In this talk we will define the braid groups of the 2-sphere and present the main differences between them and the classic braid groups of the plane, defined by Artin in 1925.
The main goal will be to describe the problem of whether or not, and under which conditions, we can consider the braid groups, on n-strands, of the 2-sphere as subgroups of braid groups, on (n+m)-strands, of the sphere.
In order to approach this question we will use the existence of the short exact sequences which involve the 2-sphere braid groups.
To conclude we will present the up-to-date answers to the above mentioned question and outline the main methods that have been used to provide these solutions.