This talk will review recent results on the construction of U(1) duality-invariant nonlinear models for gauge (2n-1)-forms in d = 4n dimensions, including $T \bar T$-like flows in the space of such theories. In the four-dimensional case, we will briefly discuss the following U(1) duality-invariant nonlinear systems: (i) models for N-extended superconformal higher-spin multiplets; (ii) the low-energy effective action for N = 4 SYM on its Coulomb branch; and (iii) models for spontaneously broken local N=1 supersymmetry. If time permits, a new formulation for a self-interacting chiral gauge 2n-form in d = 4n + 2 dimensions will be discussed.

Models with spontaneously broken local supersymmetry are naturally obtained by coupling the off-shell supergravity-matter systems to Goldstino superfields. Every irreducible Goldstino superfield produces a universal positive contribution to the cosmological constant. This talk will review the structure of N=1 and N=2 Goldstino superfields.

Based on the results of 0906.4393 and 0910.5771, this talk will discuss the formulation of general 4D N=2 superconformal sigma-model in N=2 and N=1 superspace settings.

Among the very first examples of hyper-Kahler manifolds given by Calabi in 1979, there were the cotangent bundles of complex projective spaces, T-star-CPn. Later on, many more examples of hyper-Kahler metrics on cotangent bundles of Kahler manifolds were shown to exist. Finally, Kaledin (1997) and Feix (1999) proved that a real-analytic Kahler metric on a complex manifold M can always be extended to a hyper-Kahler metric in a neighborhood of M in T-star-M. Although these mathematical proofs are rather technical and involved, there exists a streamlined physical construction which leads to the same results and is based on the concept of supersymmetry. As is well-known, four- and five-dimensional N = 2 supersymmetric nonlinear sigma-models possess the property that their target spaces are hyper-Kahler manifolds. The physical construction consists of providing a manifestly N = 2 supersymmetric nonlinear sigma-model whose target space can be shown to be (a neighborhood of the zero section in) the cotangent bundle T-star-M of a Kahler manifold M. This talk will review the salient properties of such supersymmetric nonlinear sigma-models with eight supercharges.