Recent progress in the Langlands programm provides a significant step towards the understanding of the arithmetic of global fields. The geometric Langlands program provides a systematic way to construct l-adic sheaves (resp. D-modules) on algebraic curves which subsumes the construction of classical sheaves, like rigid local systems, used in inverse Galois theory (by Belyi, Malle, Matzat, Thompson, Dettweiler, Reiter) for the construction of field extension of the rational function fields $\mathbb F_p(t)$ or $\mathbb Q(t)$ (recent work of Heinloth, Ngo, Yun and Yun). On the other hand, using Langlands correspondence for the field $\mathbb Q$, Khare, Larsen and Savin constructed many new automorphic representations which lead to new Galois realizations for classical and exceptional groups over $\mathbb Q$. It was the aim of the workshop, to bring together the experts working in the fields of Langlands correspondence and constructive Galois theory.