This Ph.D. thesis presents my results obtained in the last three years. These results have appeared in the following preprints and articles: [Pag19b, Pag19d, CDD+18, Pag18a, Pag18c, Pag19c, Pag18b, Pag19a, PP19b]. Initially, I investigated toric arrangements, a type of arrangements inspired by the hyperplane ones. Toric arrangements have been studied intensively in the last fifteen years both from topological and from combinatorial point of view. My results describe the cohomology ring of toric arrangements and their dependency from the combinatorial data. Later I have worked on another type of arrangements, i.e. elliptic arrangements, which are tougher than the toric case. I focused on the most regular case, i.e. the braid arrangements, that coincides with the configuration spaces of points in an elliptic curve. I have obtained some results on the unordered configuration space of points in an elliptic curve, and I have generalized some of them to configurations on closed orientable surfaces. Only very recently I have made some conjectures about the cohomology of braid elliptic arrangements.