  We define and study extensions of Artin's representation and braid monodromy
representation to the case of topological and algebraical generalisations of
braid groups. In particular we provide faithful representations of braid groups
of oriented surfaces with boundary components as (outer) automorphisms of free
groups. We give also similar representations for braid groups of non oriented
surfaces with boundary components and we show a representation of braid groups
of closed surfaces as outer automorphisms of free groups. Finally, we provide
faithful representations of Artin-Tits groups of type $\mathcal{D}$ as
automorphisms of free groups.
