### Abstract

I will give a survey on additive polynomials, their role in valuation theory and algebraic geometry and what we know and do not know about them. I show their connection with the defect of valued field extensions and the meaning of the defect for the model theory of valued fields and for local uniformization in positive (residue) characteristic. I will mention a classification of Artin-Schreier extensions and its applications. If time permits, I will also talk about maximality properties of valued fields and Ershov's notion of extremal fields.