indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
Abstract model theory is the study of the general properties of the model theory of extensions of (classical untyped) first-order logic.
Originally motivated by Lindström's theorem that characterizes first-order logic, the field has subsequently been extended to provide alternative characterizations and include different logics within its range.
The basic concept of abstract model theory is that of an abstract logic which is a triple $\mathcal{L}=(S,\Phi ,\models)$ where $\models$ is a binary relation between the class of $\mathcal{L}$-‘structures’ $S$ and the class of $\mathcal{L}$-‘sentences’ $\Phi$ to be thought of as minimalistic version of the satisfaction relation.
As the logical relations studied by abstract model theory are of a functorial nature, some category theory entered the picture already in Barwise (1974). The theory of institutions, aka institution-independent model theory (Diaconescu 2008), constitutes abstract categorical model theory proper. In a similar abstract categorical vein is the functorial approach to geometric theories described in Johnstone (2002, sec. B4.2).
Jon Barwise, Axioms for abstract model theory , Annals of Mathematical Logic 7 pp.221-265, 1974.
Barwise, Feferman (eds.), Model-theoretic Logics , Springer Heidelberg 1985 (freely available online: toc) .
Răzvan Diaconescu, Institution-independent Model Theory , Birkhäuser Basel 2008.
Marta García-Matos, Jouko Väänänen, Abstract Model Theory as a Framework for Universal Logic , Logica Universalis 2005 pp.1-33 (draft) .
Peter Johnstone, Sketches of an Elephant vol. I , Oxford UP 2002.
Jouko Väänänen, The Craig Interpolation Theorem and abstract model theory , Synthese 164:401 (2008) (freely available online)