What is a Rees algebra?

In this talk, we introduce the notion of the Rees algebra of an ideal, and the various ways this ring can be realized. In short, the Rees ring encapsulates the data of every power of an ideal, all within a single algebra. Hence, for algebraists, this object is invaluable within the study of multiplicities, reductions, and integral closures, to name a few. For geometers, the Rees ring serves as the coordinate ring of the graph of a rational map between varieties. As these constructions are parametric in nature, one question is how to achieve the implicitization of these algebras, namely how to obtain a tangible description of the Rees ring as a quotient of a polynomial ring. We discuss various results in this direction, as well as their applications. We end with some open questions, particularly in the much less explored realm of Rees rings of modules.