Abstract: Last week I mentioned that the quantum minors contained in a given H-invariant prime ideal of quantum matrices form a Grobner basis for the ideal. I will give an outline of this proof on May 6th. In preparation for this, I will give a brief introduction to Grobner bases this week. These objects, while only developed in the 1960's as a generalization of both the Euclidean algorithm and Gaussian row reduction, have since formed a fundamental tool in commutative algebra and algebraic geometry. I will introduce their basic theory and applications. I will then discuss extensions of the theory to certain noncommutative algebras such as, of course, quantum matrices.