 The complementary approaches, rigor and pragmatics, which are exhibited in the development of quantum mechanics, later came about in a more striking way in connection with the development of quantum electrodynamics (QED) and, more generally, quantum field theory (QFT). The emphasis on rigor emerges in connection with two frameworks, algebraic QFT and Wightman’s axiomatic QFT. Algebraic QFT has its roots in the work of von Neumann on operator algebras, which was developed by him in an attempt to generalize the Hilbert space framework. Wightman’s axiomatic QFT has its roots in Schwartz’s theory of distributions, and it was later developed in the rigged Hilbert space framework. Roughly, the basic distinction between the two approaches is that the algebra of operators is the basic mathematical concept in algebraic QFT, while operator-valued distributions (the quantum analogues of field quantities) are fundamental in Wightman’s axiomatic QFT. It is worth noting that algebraic QFT is generally formulated axiomatically, and that it is just as deserving of the name “axiomatic” QFT. However, that term is often taken to refer specifically to the approach based on operator-valued distributions. To avoid any possible confusion, that approach is referred to here as “Wightman’s axiomatic” QFT. The emphasis on pragmatics arises most notably in Lagrangian QFT, which uses perturbation theory, path integrals, and renormalization techniques. Although some elements of the theory were eventually placed on a firmer mathematical foundation, there are still serious questions about its being a fully rigorous approach on a par with algebraic and Wightman’s axiomatic QFT. Nevertheless, it has been spectacularly successful in providing numerical results that are exceptionally accurate with respect to experimentally determined quantities, and in making possible expedient calculations that are unrivaled by other approaches.