  A henselian valued field $K$ is called separably tame if its
separable-algebraic closure $K^{\operatorname{sep}}$ is a tame extension, that
is, the ramification field of the normal extension $K^{\operatorname{sep}}|K$
is separable-algebraically closed. Every separable-algebraically maximal
Kaplansky field is a separably tame field, but not conversely. In this paper,
we prove Ax-Kochen-Ershov Principles for separably tame fields. This leads to
model completeness and completeness results relative to the value group and
residue field. As the maximal immediate extensions of separably tame fields are
in general not 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. Our approach also yields
alternate proofs of known results for separably closed valued fields.
