 Most physicists use Lagrangian QFT (LQFT) to make predictions that have been experimentally verified with extraordinary precision in some cases. However, LQFT has been described as a “grab bag of conflicting mathematical ideas” that has not provided a sharp mathematical description of what counts as a QFT model (Swanson 2017, pp. 1–2). Those criticisms motivated mathematically inclined physicists to search for a mathematically rigorous formulation of QFT. Axiomatic versions of QFT have been favored by mathematical physicists and most philosophers. With greater mathematical rigor it is possible to prove results about the theoretical structure of QFT independent of any particular Lagrangian. Axiomatic QFT provides clear conceptual frameworks within which precise questions and answers to interpretational issues can be formulated. There are three main axiomatic frameworks for QFT: Wightman QFT, Osterwalder-Schrader QFT, and algebraic QFT. In Wightman QFT, the axioms use functional analysis and operator algebras and is closer to LQFT since its axioms describe covariant field operators acting on a fixed Hilbert space. The Osterwalder-Schrader axioms use a functional integration approach to QFT. The algebraic QFT axioms use \(C^*\)-algebras to model local observables. However, axiomatic QFT approaches are sorely lacking with regards to building empirically adequate models. Unlike quantum mechanics which has a canonical mathematical framework in terms of von Neumann’s Hilbert space formulation, QFT has no canonical mathematical framework. Even though there is a canonical mathematical framework for quantum mechanics, there are many interpretations of that framework, e.g., many-worlds, GRW, Copenhagen, Bohmian, etc... QFT has two levels that require interpretation: (1) which QFT framework should be the focus of these foundational efforts, if any, and (2) how that preferred framework should be interpreted. Since (1) involves issues about mathematical rigor and pragmatic virtues, it directly bears on the focus of this article. The lack of a canonical formulation of QFT threatens to impede any metaphysical or epistemological lessons that might be learned from QFT.