Abstract : The braid group on $n$ strands on a surface $S$ is a natural extension of both the classical braid group $B_n$ and the fundamental group of $S$. In the first part of this thesis, we give new presentations for surface braid groups, improving presentations earlier obtained by Scott and González-Meneses. Afterwards we show how to associate to any graph with $n$ vertices on the sphere a presentation for the braid group on $n$ strands on the sphere, generalising Sergiescu's result on planar graphs. We determine also the Outer group of braid groups on the sphere. Afterwards we generalise Fenn-Rolfsen-Zhu results on centralisers in classical braid groups to surface braids. As consequence of this result, we prove that the word problem for the monoids of singular surface braids is solvable. In the last part of the thesis we study cubic Hecke algebras. Extending Jones' approach, we prove the existence of a Markov trace on a suitable tower of quotients of these algebras. We obtained in this way two new link invariants that are recursively computable, different from the HOMFLY and Kauffman polynomials, and defined by two explicit skein relations, where one of them is cubic.