In this tutorial we introduce the mathematical notion of generic figures as a way to formalize logical diagrammatic notation and forms of diagrammatic reasoning. Generic figures are constitutive components of presheaf categories, which are themselves standard constructions in the mathematics of category theory. Using presheaves and generic figures, it is possible to capture in an intuitive and natural way the relevant structural properties of certain diagrammatic logical systems, in particular the Existential Graphs developed by C. S. Peirce. Not only are the distinct parts and compositional properties of logical diagrams readily represented with generic figures, but the comparative and transformational processes linking and distinguishing diagrams in such systems with respect to one another, thus serving as the basis for reasoning with them, are also susceptible to formalization using the same mathematical tools. This tutorial aims to make these widely applicable techniques available to a broad audience without presuming any background in category theory or previous familiarity with Peirce’s system. All of the mathematical understanding necessary in order to understand presheaves, generic figures, and their application to Peirce’s Existential Graphs will be presented in a self-contained manner.