Commutative Algebra, flushed with its great successes in dealing with the difficult homological questions that originated in the 60's, has flourished recently around many clusters of related activities. To name a few, in no particular order of significance: linkage theory, Cohen–Macaulay approximation, theory of Hilbert functions, the structure of free resolutions, tight closure, and a growing web of relationships to computer algebra and combinatorics. All this, while strengthening its connections to algebraic geometry and other parts of algebra and retaining the vitality of its established domains.

This is a panorama that begs for systematic accounts of these developments to guide a possible reader through a bountiful but scattered literature. Several efforts have begun to address these needs and there are definite stirrings of an accelerating pace.

Another of these nuclei of activity is the theory and practice of the so-called Rees or blowup algebras. Through its many variants it interacts in some measure with all the areas listed above. Furthermore, it is generating questions at a much faster clip than they are being dealt with. It became then a likely target to be written about.

We decided to assemble some recent results and methods on these algebras in a manner which would be comprehensible/useful to a reader conversant with basic commutative algebra. It was the original intent to frame these notes into a second course in commutative algebra, requiring the core part of [192], along with a good foundation in homological algebra.