Differential categories are now a well-studied abstract setting for differentiation. However not much attention has been given to the process which is inverse to differentiation: integration. This thesis presents an analogous study of integral categories. Integral categories give an abstraction of integration by axiomatizing extra structure on a symmetric monoidal categories with a coalgebra modality using the primary rules of integration. The axioms for integrations include the analogues of integration by parts rule, also called the Rota-Baxter rule, the independence of the order of iterated integrals and that integral of any constant map is linear. We expect consequences of the compatible interaction between integration and differentiation to include the two fundamental theorems of calculus. A differential category with integration which satisfies these two theorem in a suitable sense is what we call a calculus category.