The question whether a compact periodic group has a finite exponent has been known since early 60's. One can find a discussion on this question in the book on Abstract Harmonic Analysis by Hewitt and Ross. Recently, Zelmanov, developing his ideas on the solution of the restricted Burnside problem, has proved that compact periodic groups are locally finite. Hence, every Sylow p-subgroup of compact periodic groups are Engel. I will give an outline of an approach which leads to a necessary and sufficient condition for a periodic compact group to be of finite exponent. It turned out that a periodic compact group has a finite exponent if and only if it is strongly Engel. As usual, we will talk about open problems in this area.