From the twilight of cultural memory: the 'Bumah' in the mosques of central Oman

We give a short introduction to the theory of twisted Alexander polynomials of a 3--manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications of twisted invariants to the study of topological problems. We conclude with a short summary of the theory of higher order Alexander polynomials.Comment: 42 pages, final version of the survey paper to be published by the proceedings of the conference `The Mathematics of Knots: Theory and Application' in Heidelberg in December 2008. In the final version we updated references and fixed a few typo