The theory of blowup algebras Rees algebras, associated graded rings, Hilbert functions, and birational morphisms — is undergoing a period of rapid development. One of the aims of this book is to provide an introduction to these developments. The emphasis is on deriving properties of rings from their specifications in terms of generators and relations. While this places limitations on the generality of many results, it opens the way for the application of computational methods. A highlight of the book is the chapter on advanced computational methods in algebra built on current understanding of Gr?bner basis theory and advanced commutative algebra. In a concise way, the author presents the Gr?bner basis algorithm and shows how it can be used to resolve many computational questions in algebra.