When von Neumann introduced the concept of amenable groups in 1929, he showed that no group containing a nonabelian free subgroup was amenable. The von Neumann-Day conjecture was a converse to this, that any nonamenable group must contain F2 as a subgroup. This was originally disproved by Ol'shanskii in 1980, and since then a wide range of counterexamples have emerged. One may then ask, are there any large classes of group where the von Neumann-Day conjecture holds? Or, does some similar statement hold if we weaken what is meant by having a nonamenable free subgroup? In this talk, I will discuss some positive answers to the first question, in particular groups which satisfy a Tits alternative. Then, I will discuss progress towards the second in the field of measured group theory, and the Gaboriau-Lyons problem.