Although the title of this book is “Computational Algebraic Geometry”, it could also be titled “Snapshots of Commutative Algebra via Macaulay 2”. The aim is to bring algebra, geometry, and combinatorics to life by examining the interplay between these areas; it also provides the reader with a taste of algebra different from the usual beginning graduate student diet of groups and field theory. As background the prerequisite is a decent grounding in abstract algebra at the level of [56]; familiarity with some topology and complex analysis would be nice but is not indispensable. The snapshots which are included here come from commutative algebra, algebraic geometry, algebraic topology, and algebraic combinatorics. All are set against a backdrop of homological algebra. There are several reasons for this: first and foremost, homological algebra is the common thread which ties everything together. The second reason is that many computational techniques involve homological algebra in a fundamental way; for example, a recurring motif is the idea of replacing a complicated object with a sequence of simple objects. The last reason is personal – I wanted to give the staid and abstract constructs of homological algebra (e.g. derived functors) a chance to get out and strut their stuff. This is said only half jokingly – in the first class I ever had in homological algebra, I asked the professor what good Tor was; the answer that Tor is the derived functor of tensor product did not grip me.