This will be an informal talk summarizing roughly where we stand concerning the problem of local uniformization in positive characteristic. I will discuss the structure of valued algebraic function fields and the main open problems that are of particular interest for local uniformization. These include the dehenselization problem (an analogue of Temkin’s “decompletion”) as well as the question when the existence of a rational place implies that the ground field is existentially closed in the function field.

If time permits I will also discuss a stunning result about the badness of valuations in positive characteristic (due to Anna Blaszczok), and state a result (by Anna and myself) and an open question that are of interest for recent work by Koen Struyve et al. on “Euclidean buildings.”