et k be a field of characteristic zero, K an algebraic function field over k, and V a k-valuation ring of K. Zariski's theorem of local uniformization shows that there exist algebraic regular local rings with quotient field K which are dominated by V, and such that the direct limit .

A fter giving the necessary background and definitions we discuss generalizations of Zariski's theorem and we give examples of valuations that arise naturally in Algebraic Geometry.