Gauge $p$-forms in diverse dimensions are ubiquitous in supergravity and string theory. This work reviews novel covariant formulations designed to generate arbitrary interacting duality-invariant or chiral (self-dual) $p$-form theories in $d = 2p + 2$ space-time dimensions. For odd $p$, such theories possess $\mathsf{U}(1)$ duality invariance and include the Born-Infeld and ModMax theories. For even $p$, they describe a self-interacting chiral $p$-form with its gauge-invariant field strength obeying a nonlinear self-duality condition. We provide a complete description of $T\bar T$-like deformations of $\mathsf{U}(1)$ duality-invariant models for nonlinear electrodynamics in four dimensions and their six-dimensional counterparts -- interacting chiral two-form field theories. We also elaborate on consistent flows in the spaces of duality-invariant or chiral (self-dual) $p$-form theories beyond six dimensions.

12 pages, based on the talk given at the XVIth Quark Confinement and the Hadron Spectrum Conference (QCHSC24), 19-24 August, 2024, Cairns, Queensland, Australia