Allegories abstract and generalise (in the categorical framework) the algebra of binary relations. Arrows in an allegory enjoy a lot of properties and structure available for plain binary relations. At the same time, allegories are sufficiently general to allow the description within the same uniform framework of the lattice valued (e.g., fuzzy) relations and some more general structures. The paper presents a conceptual data modelling formalism which uses the language of allegories. We will provide examples demonstrating expressiveness of this formalism. While most of the examples are meant to be interpreted in the allegory of sets and binary relations, we also show the usefulness of using other allegories, such as the allegory of sets and lattice valued relations, with which one can model replicated data or data stored in a valid time temporal database.