The theory of Gro?bner bases, invented by Bruno Buchberger, is a general method by which many fundamental problems in various branches of mathematics and engineering can be solved by structurally simple algorithms. The method is now available in all major mathematical software systems. This book provides a short and easy-to-read account of the theory of Gro?bner bases and its applications. It is in two parts, the first consisting of tutorial lectures, beginning with a general introduction. The subject is then developed in a further twelve tutorials, written by leading experts, on the application of Gro?bner bases in various fields of mathematics. In the second part are seventeen original research papers on Gro?bner bases. An appendix contains the English translations of the original German papers of Bruno Buchberger in which Gro?bner bases were introduced.