This article by N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček explores two constructions of hyperkähler manifolds: one based on a Legendre transform and the other on a symplectic quotient. These constructions initially emerged in the context of supersymmetric nonlinear \(\sigma\)-models but are described geometrically. The authors aim to clarify the relationship between supersymmetry and modern differential geometry, making the concepts accessible to a broader audience. The article covers foundational topics such as superalgebras, superfields, and supersymmetry transformations, and provides detailed examples and interpretations of the constructions. It also reviews essential background material, including Kähler and hyperkähler geometry, quotients, symplectic and Kähler quotients, twistor theory, and nonlinear \(\sigma\)-models. The constructions are derived using supersymmetry, and the article includes references to relevant reviews for physicists.This article by N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček explores two constructions of hyperkähler manifolds: one based on a Legendre transform and the other on a symplectic quotient. These constructions initially emerged in the context of supersymmetric nonlinear \(\sigma\)-models but are described geometrically. The authors aim to clarify the relationship between supersymmetry and modern differential geometry, making the concepts accessible to a broader audience. The article covers foundational topics such as superalgebras, superfields, and supersymmetry transformations, and provides detailed examples and interpretations of the constructions. It also reviews essential background material, including Kähler and hyperkähler geometry, quotients, symplectic and Kähler quotients, twistor theory, and nonlinear \(\sigma\)-models. The constructions are derived using supersymmetry, and the article includes references to relevant reviews for physicists.