The algorithm of long division, first for integers and then polynomials, facilitated decisions regarding factors and greatest common divisors. This led to the idea of Euclidean domains, which in turn opened up new rings in which division algorithms could play a role. With the onset of computing power and the breakthrough results of Bruno Buchberger, the notion of Gröbner bases has come into prominence as a tool for managing the intricacies of polynomials in several variables and computational problems in commutative algebra. The time is right for a theory of Gröbner bases that encompasses the aforementioned algorithmic structures. The present book does this by focusing on the notion of a Gröbner function from a module into a well-partially ordered set. An abstract theory emerges in which new modules and domains can have Gröbner bases, and wherein new questions can be posed.