Aspect-Oriented Programming (AOP) started ten years ago with the remark that modularization of so-called crosscutting functionalities is a fundamental problem for the engineering of large-scale applications. Originating at Xerox PARC, this observation has sparked the development of a new style of programming featured that is gradually gaining traction. However, AOP lacks theoretical foundations to clarify new ideas showing up in its wake. This paper proposes to put a bridge between AOP and the notion of 2-category to enhance the conceptual understanding of AOP. Starting from the connection between the λ-calculus and the theory of categories, we provide an internal language for 2-categories and show how it can be used to define the first categorical semantics for a realistic functional AOP language, called MinAML. We then take advantage of this new categorical framework to introduce the notion of computational 2-monads for AOP. We illustrate their conceptual power by defining a 2-monad for Éric Tanter's execution levels--which constitutes the first algebraic semantics for execution levels--and then introducing the first exception monad transformer specific to AOP that gives rise to a non-flat semantics for exceptions by taking levels into account.