  A henselian valued field $K$ is called a tame field if its algebraic closure
$\tilde{K}$ is a tame extension, that is, the ramification field of the normal
extension $\tilde{K}|K$ is algebraically closed. Every algebraically maximal
Kaplansky field is a tame field, but not conversely. We develop the algebraic
theory of tame fields and then prove Ax--Kochen--Ershov Principles for tame
fields. This leads to model completeness and completeness results relative to
value group and residue field. As the maximal immediate extensions of tame
fields will in general not be unique, the proofs have to use much deeper
valuation theoretical results than those for other classes of valued fields
which have already been shown to satisfy Ax--Kochen--Ershov Principles. The
results of this paper have been applied to gain insight in the Zariski space of
places of an algebraic function field, and in the model theory of large fields.
