Homotopy theories for braids and topological generalizations

In this thesis we consider braid groups up to homotopy. Recently it was found a new presentation for homotopy braid groups, more symmetric than the classical one, and implying that the pure braid group quotiented by the homotopy relation is isomorphic to the reduced pure braid group. This presentation was found using a topologic approach. In this work we find the same presentation by a combinatorial approach. Then we focus on surface braids and welded braids, giving an overview of the groups and of the results existing in the literature regarding their homotopy.