The last chapter provides the background knowledge needed to begin our study of first-order theories and their models. In Sect. 3.1 we discuss them from an abstract perspective, in order to describe a fundamental correspondence between sets of sentences in a first-order language and the corresponding classes of first-order structures (this correspondence essentially amounts to what is known as a Galois connectionGalois connection). Sections 3.2 and 3.3 introduces some of the algebraic theories that will feature prominently in the rest of the book. More precisely, in Sect. 3.2 we axiomatise some special classes of abelian groups as well as vector spaces, whose dimension theory we outline for future reference. In Sect. 3.3 we axiomatise rings and fields but focus essentially on rings, and especially rings of polynomials, which will play a very important role in the second half of the book. Special properties of polynomial rings, which will be used throughout Chap. 10 , are proved in the Addendum.