Should I begin by defining ‘model theory’? This might be unsafe – do the readers get their money back if the definition doesn't match the contents? But here is an attempt at a definition: Model theory is the study of the construction and classification of structures within specified classes of structures.

A ‘specified class of structures’ is any class of structures that a mathematician might choose to name. For example it might be the class of abelian groups, or of Banach algebras, or sets with groups which act on them primitively. Thirty or forty years ago the founding fathers of model theory were particularly interested in classes specified by some set of axioms in first-order predicate logic – this would include the abelian groups but not the Banach algebras or the primitive groups. Today we have more catholic tastes, though many of our techniques work best on the first-order axiomatisable classes. One result of this broadening is that model theorists are usually much less interested than they used to be in the syntactical niceties of formal languages – if you want to know about formal languages today, you should go first to a computer scientist.

‘Construction’ means building structures, or sometimes families of structures, which have some feature that interests us.