1. To give a leisurely and fairly comprehensive introduction to the definition and construction of Grobner bases;
2. To discuss applications of Grobner bases by presenting computational methods to solve problems which involve rings of polynomials.

This book is designed to be a first course in the theory of Grobner bases suitable for an advanced undergraduate or a beginning graduate student. This book is also suitable for students of computer science, applied mathematics, and engineering who have some acquaintance with modern algebra. The book does not assume an extensive knowledge of algebra. Indeed, one of the attributes of this subject is that it is very accessible. In fact, all that is required is the notion of the ring of polynomials in several variables (and rings in general in a few places, in particular in Chapter 4) together with the ideals in this ring and the concepts of a quotient ring and of a vector space introduced at the level of an undergraduate abstract and linear algebra course. Except for linear algebra, even these ideas are reviewed in the text. Some topics in the later sections of Chapters 2, 3, and 4 require more advanced material. This is always clearly stated at the beginning of the section and references are given. Moreover, most of this material is reviewed and basic theorems are stated without proofs.