The quote from Alfred Tarski in Sect. 1.5 described model theory as a study of the relation between mathematical systems and the sentences that hold in them. Tarski’s description sets the key tasks of this chapter: we shall define mathematical systems (or structures, as we shall call them), the linguistic resources we are to employ, and a relation of ‘satisfaction’, to spell out, in mathematically useful terms, what we mean when we say that a particular statement ‘holds’ in a particular system or structure. In Sects. 2.1 and 2.2 we specify our linguistic resources. In particular, we explicitly declare which symbol sets or alphabets we use and describe how we concatenate symbols to form expressions belonging to our syntactic categories, i.e. terms and formulae. The next task is to establish a systematic correspondence between formulae and certain set-theoretic objects, which we refer to as structures. This task is carried out in Sect. 2.3, where the notion of a structure, which is relative to a fixed language, is introduced and its connection with a formal language is given by defining the satisfaction relation. Satisfaction holds between a formula \(\varphi \) and a structure \(\mathcal {M}\) when \(\varphi \) is true (in a sense to be specified) in \(\mathcal {M}\) , possibly after the variables occurring in \(\varphi \) have been assigned fixed values. In Sect. 2.4 we use satisfaction to define the fundamental relation of consequence between two formulae or, more generally, between a set of formulae and a formula. Section 2.5 is devoted to proofs by induction on the complexity of formulae, which yield some abstract theorems about formulae and provide an opportunity to practice a proof technique that will often be appealed to in later chapters. The last section, Sect. 2.6, contains a succinct discussion of definability and interpretations.