The Burnside problem asks whether it is possible for a finitely generated group to be infinite if all its elements have finite order. This version is usually called the General Burnside problem and an example of such a group was found in 1964.
The Burnside group B(d, n) is the largest d generator group in which every element satisfies xn = 1.
The Restricted Burnside problem asks whether, for fixed d and n, there is a largest finite d generator group in which every element satisfies xn = 1. A positive solution to the Restricted Burnside problem would show that there are only finitely many finite factor groups of B(d, n).