Ever since Gromov's seminal work, the hard question of understanding the possible symplectic images of a Euclidean ball has been a driving question in symplectic topology. In this talk, I will report on some recent results in this direction, with particular emphasis on the so-called Lagrangian products, which are certain subsets of the standard phase space. It turns out that classical integrable systems and billiards play an important role in this story. This is joint work with Y. Ostrover and V. G. B. Ramos.