Studying first-order theories via combinatorial properties of formulae lies at the heart of pure model theory. The question whether these so-called dividing lines correspond to purely algebraic notions when applied to fields has been a driving force in the model theory of fields for the past 40 years. In this talk, we introduce the relevant model-theoretic combinatorial definitions (including NIP and dp-rank) as well as discuss their field-theoretic consequences. In particular, we give an overview of the area and its main conjectures.
