  We prove that every place of an algebraic function field F|K of arbitrary
characteristic admits local uniformization in a finite extension F' of F. We
show that F'|F can be chosen to be normal. If K is perfect and P is of rank 1,
then alternatively, F' can be obtained from F by at most two Galois extensions;
if in addition P is zero-dimensional, then we only need one Galois extension.
Certain rational places of rank 1 can be uniformized already on F. We introduce
the notion of "relative uniformization" for arbitrary finitely generated
extensions of valued fields. Our proofs are based solely on valuation
theoretical theorems, which are of fundamental importance in positive
characteristic.
