The physics of non-axisymmetry is a far more important topic in the theory of toroidal fusion plasmas than might be expected. (1) Even a small toroidal asymmetry in the magnetic field strength, δ ≡ ∂ln B/∂φ ∼ 10−4, can cause an unacceptable degradation in performance. (2) Nevertheless, asymmetries—even large asymmetries δ ∼ 1—can give beneficial plasma control and circumvent issues, such as magnetic-configuration maintenance and plasma disruptions, that make axisymmetric fusion devices problematic. Viewed from prospectives that are adequate for designing and studying axisymmetric plasmas, the physics of non-axisymmetric plasmas appears dauntingly difficult. Remarkably, Maxwell's equations provide such strong constraints on the physics of toroidal fusion plasmas that even a black-box model of a plasma answers many important questions. Kinetic theory and non-equilibrium thermodynamics provide further, but more nuanced, constraints. This paper is organized so these constraints can be used as a basis for the innovations and for the extrapolations that are required to go from existing experiments to fusion systems. Outlines are given of a number of calculations that would be of great importance to ITER and to the overall fusion program and that could be carried out now with limited resources.