Computing power integral bases in algebraic number fields

It is a classical problem of algebraic number theory to decide if the ring of integers of an algebraic number field admits power integral bases and to determine all generators of power integral bases. The purpose of the present paper is to give a survey on the algorithms developed in the recent couple of years to compute generators of power integral bases in algebraic number fields. These algorithms provide general, fast methods for lower degree fields, and also several efficient methods for certain classes of higher degree fields.