Øystein Ore proved in $1951$ that for $n\gt4$ we have that all elements of $A_n$ are commutators.  He subsequently conjectured that this is true for all finite non-abelian simple groups.

The theorem was proved in $2008,$ and relies on the *classification theorem*.  

 - See M. W. Libeck, E. A. O'Brien, 
Aner Shalev, Shame Huu Tiep,  *Journal of the European Mathematical Society*, vol. $12$, issue $4, 2010.$  pp. $939-1008.$

Interestingly Ore himself was mainly concerned with graph theory.