An introduction to the universal algebra approach to Higman-Thompson groups (including Thompson's group V ) is given, following a series of lectures by Graham Higman in 1973. In these talks, Higman outlined an algorithm for the conjugacy problem; which although essentially correct fails in certain cases, as we show here. A revised and complete version of the algorithm is written out explicitly. From this, we construct an algorithm for the power conjugacy problem in these groups. Python implementations of these algorithms can be found at [26].