In this talk, I will discuss some questions related to the nonlinear theory of electromagnetism formulated by Born and Infeld in 1934. I will discuss the link between this theory and the curvature operators in the Euclidean and in the Lorentz-Minkowski space. I will address the solvability of the electrostatic Born-Infeld equation with sources (and a model driven by the nonlinear Klein-Gordon equation) emphasizing the open questions and the partial recent progress we made. Finally, I will present some more academic results related to the curvature operator in the Lorentz-Minkowski space and the solvability of some scalar field type equations.