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Motivated by the quest for a basic description of the six-dimensional (2,0)-theory, as well as the constructions of Yang-Mills theories on a noncommutative space (NCYM) by Douglas and Hull and by Connes, Douglas, and Schwarz, the "curvepole" deformation was introduced two decades ago. This is a novel kind of quantum field theory, which (like NCYM) includes nonlocal interactions in a controlled way. Two decades ago, it was indirectly defined as the (by conjecture, decoupled) 6d sector of the low-energy limit of an M-theory setup. The simplest version is an interacting deformation of the free 6d tensor multiplet. More recently, a (partial) construction of an action was put forward, by combining supersymmetry with a certain "invariance" principle that will be described in the talk. The action features an interacting antiselfdual 2-form gauge field, and it predicts new kinds of interactions of M5-brane fields and background 4-form flux (of M-theory). Curvepole theories may also offer new kinds of UV-completions for nonrenormalizable 5d gauge theories. I will discuss this partial progress as well as remaining challenges.

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