We study the valuation theory of deeply ramified fields and introduce and
investigate several other related classes of valued fields. Further, a
classification of defect extensions of prime degree of valued fields that was
earlier given only for the characteristic equal case is generalized to the case
of mixed characteristic. It is shown that deeply ramified fields and the other
valued fields we introduce only admit one of the two types of defect
extensions, namely the ones that appear to be more harmless in open problems
such as local uniformization and the model theory of valued fields in positive
characteristic. The classes of valued fields under consideration can be seen as
generalizations of the class of tame valued fields. The present paper supports
the hope that it will be possible to generalize to deeply ramified fields
several important results that have been proven for tame fields and were at the
core of partial solutions of the two mentioned problems.