In 1902, William Burnside asked whether a finitely generated group in which every element has finite order must be finite. This question, now known as the Burnside Problem, was answered negatively by Novikov and Adian in 1968, who proved that the free Burnside group B(m,n) is infinite for sufficiently large odd exponents n. More recently, in 2023, Agatha Atkarskaya, Eliyahu Rips, and Katrin Tent proved that B(m,n) is infinite for all odd n?557, the best currently known bound, using methods from iterated small cancellation theory. After outlining the ideas behind this result, we discuss connections between free Burnside groups, the space of amenable subgroups, and C*-simplicity, including work in progress with Katrin Tent and Sam Shepherd on constructing new classes of C*-simple groups via Burnside quotients.