We discuss sympletic, contact and locally conformal symplectic structures. We show how they are connected and how they organize themselves inside the category of Jacobi structures. An emphasis in put on the role of their automorphism groups since they encode the corresponding geometry in the spirit of the Erlanger Programme. We also evoque the problems of existence and (local and global) classification of the structures. For the background of the material discussed in this article, see [30].