Positive monotone symplectic manifolds are the analogue of Fano varieties in symplectic geometry. In the presence of a Hamiltonian torus action it is in some cases possible to prove boundedness results on positive monotone manifolds which hold for smooth complex projective Fano varieties. In this talk I will report on such results, some of them are joint work with Leonor Godinho and Silvia Sabatini, some of them are joint work with Dmitri Panov. Then I will discuss some further questions and work in progress.