Noncommutative Algebraic Geometry: from pi-algebras to quantum groups

The main purpose of this paper is to provide a survey of different notions of algebraic geometry, which one may associate to an arbitrary noncommutative ring R. In the first part, we will mainly deal with the prime spectrum of R, endowed both with the Zariski topology and the stable topology. In the second part we focus on quantum groups and, in particular, on schematic algebras and show how a noncommutative site may be associated to the latter. In the last part, we concentrate on regular algebras, and present a rather complete up to date overview of their main properties.