Infinite torsion group with finitely many conjugacy classes Do there exist infinite torsion groups with finitely many conjugacy classes? One can easily see that there are no such groups with only two conjugacy classes. Note also that one can construct torsion-free groups with finitely many conjugacy classes via HNN extensions.

Yes there are groups like this. In Geometry of Defining Relations in Groups by Ol'shanskii this is described in the last chapter, and as a particular example he gives infinite $p$ groups with exactly $p$ conjugacy classes (Thm 41.2), and I think he give credit to Ivanov for the basic approach.
He uses the same small cancellation technology used to construct Tarski monsters (fg infinite p-groups).
