Abstract: The talk is a report on the recent completion of a project dating from 2010. Our goal is an explicit description of the additive and multiplicative structure of the cohomology of a polyhedral product. The result, for field coefficients and general CW pairs, comes with a transparency sufficient to allow for computation. The problem has an extensive history which I shall try to outline briefly. Some of the earliest results in this direction, for the case of moment-angle complexes, appeared in the work of M. Franz, I. Baskakov, V. Buchstaber and T. Panov and also S. Lopez de Medrano. The focus in this lecture will be on the additive results. This is joint work with M. Bendersky, F.R. Cohen and S. Gitler.