Abstract: Since their introduction in the early 1960s, Anosov flows have defined an important class of dynamics, thanks to their many interesting chaotic features and rigidity properties. Moreover, their topological aspects have been deeply explored, in particular in low dimensions, thanks to the use of foliation theory in their study. Although the connection of Anosov flows to contact and symplectic geometry was noted in the mid 1990s by Mitsumatsu and Eliashberg-Thurston, such interplay has been left mostly unexplored. I will present some recent results on the contact and symplectic geometric theory of Anosov flows in dimension 3. Time permitting, various related topics will be discussed, including the interplay of Anosov flows with Reeb dynamics, Liouville geometry and surgery theory, the presence of invariant volume forms, etc.