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We prove in arbitrary characteristic that an immediate valued algebraic function field $F$ of transcendence degree 1 over a tame field $K$ is contained in the henselization of $K(x)$ for a suitably chosen $x\in F$. This eliminates ramification in such valued function fields. We give generalizations of this result, relaxing the assumption on $K$. Our theorems have important applications to local uniformization and to the model theory of valued fields in positive and mixed characteristic., new improved version

Several classes of henselian valued fields admit quantifier elimination relative to structures which reflect the additive and multiplicative congruences of the field. Value groups and residue fields may be viewed as reduts of these structures. A general theorem is given using the theory of tame extensions of henselian fields. Special cases like the case ofp-adically closed fields and the case of henselian fields of residue characteristic 0 are discussed.