  In this article, we review the proofs of the first Zassenhaus Conjecture on
conjugacy of torsion units in integral group rings for the alternating groups
of degree 5 and 6, by Luthar-Passi and Hertweck. We describe how the study of
these examples led to the development of two methods -- the HeLP method and the
lattice method. We exhibit these methods and summarize some results which were
achieved using them. We then apply these methods to the study of the first
Zassenhaus conjecture for the alternating group of degree 7 where only one
critical case remains open for a full answer. Along the way we show in examples
how recently obtained results can be combined with the methods presented and
collect open problems some of which could be attacked using these methods.
