  We prove that every place P of an algebraic function field F|K of arbitrary
characteristic admits local uniformization in a finite extension E of F. We
show that E|F can be chosen to be Galois, after a finite purely inseparable
extension of the ground field K. Instead of being Galois, the extension can
also be chosen such that the induced extension EP|FP of the residue fields is
purely inseparable and the value group of F only gets divided by the residue
characteristic. If F lies in the completion of an Abhyankar place, then no
extension of F is needed. Our proofs are based solely on valuation theoretical
theorems, which are of particular importance in positive characteristic. They
are also applicable when working over a subring R of K and yield similar
results if R is regular and of dimension smaller than 3.
