Definable Categories and Monoidal Categories

Definable categories are axiomatisable additive categories. They appear as definable subcategories of module categories, equivalently as the categories of exact functors on some small abelian category. We give an exposition of their structure and their model theory from an essentially intrinsic point of view. We recall the anti-equivalence between definable categories and small abelian categories and we describe a monoidal version of this due to Wagstaffe.