  We classify all possible extensions of a valuation from a ground field $K$ to
a rational function field in one or several variables over $K$. We determine
which value groups and residue fields can appear, and we show how to construct
extensions having these value groups and residue fields. In particular, we give
several constructions of extensions whose corresponding value group and residue
field extensions are not finitely generated. In the case of a rational function
field $K(x)$ in one variable, we consider the relative algebraic closure of $K$
in the henselization of $K(x)$ with respect to the given extension, and we show
that this can be any countably generated separable-algebraic extension of $K$.
In the "tame case", we show how to determine this relative algebraic closure.
Finally, we apply our methods to power series fields and the $p$-adics.
