We look at some recent work of Leitner and Pawloski. They give a proof of the Poincare-Birkhoff-Witt theorem using Grobner bases and suggest a Grobner basis approach to deformation quantization. We will start with the basic properties and applications (including the definition) of Grobner bases for commutative algebras over a field, and then look at the generalizations to associative algebras over arbitrary ground rings that Lietner and Pawloski introduce.