A central theme in the C*-theory is the construction of operator algebras from a set of geometric/topological data and the investigation of their properties. When the model uses nuclear building blocks and amenable algebraic methods then the C*-output should be nuclear. Then additional properties of the data should reflect to properties of the C*-output placing it in the classification programme. This scheme has been met with quite success for C*-algebras of C*-correspondences (work by Katsura, Schweizer, Brown-Tikuisis-Zelenberg, Carlsen-Kwasniewski-Ortega, and Fletcher-Gillaspy-Sims to name but a few).

In this talk I will report on recent developments about product systems. The objective is to move from the semigroup of the natural numbers (C*-correspondences) to more general semigroups (free abelian semigroup, right angled Artin semigroups, etc.). These results include characterizations of nuclearity and exactness, as well as critical values of phase transitions in the presence of finite frames.