I plan to report on the following local uniformization results (co-authors and myself) for function fields F|K: First, every place of F|K of admits local uniformization, provided it has not transcendence defect. (We call such places Abhyankar places). Since Abhyankar places lie dense in the Riemann-Zariski space of all places of F|K with respect to the patch topology, an interesting question is whether one can ``catch'' the bad places by approximatng them wih Abhyankar places in such a way that local uniformization is maintained. Second, if F|K is separable, every place admits local uniformization in a finite separable extension F' of F. This follows actually from de Jong's result; but we can show in addition that F'|F can be chosen to be Galois. Alternatively, F'|F can be chosen to satisfy a valuation theoretical condition which is very natural in positive characteristic. Our proofs are based solely on valuation theoretical theorems, which are of fundamental importance in positive characteristic. We will mention how the valuation theoretical phenomenon of defect of a valuation is responsible for making our life harder in positive charactersitic. Finally, this result of ours is in some sense ``orthogonal'' to Temkin's result, who proves local uniformization for a purely inseparable extension of the function field.