At a Dagstuhl meeting a few years ago, I gave a tutorial lecture about Groebner bases whose emphasis was on their defining properties and the classical algorithms for computing them. In this year's tutorial, I will focus more on what we can do with Groebner bases and polynomial ideal theory. The plan is to cover some of the standard situations in which Groebner bases are helpful, as well as some recent joint work with Armin Biere and Daniela Ritirc on using Groebner bases for circuit verification.