Title: Valued function fields and the defect

I will give a survey on the role that the valuation theretical phenomenon 
of "defect" (or "ramification deficiency") plays in the structure theory 
of valued function fields in positive characteristic. The defect is the 
main enemy that we meet when we deal with the following two deep problems 
in positive (or mixed) characteristic:
- the model theory of valued fields, in particular that of Laurent series 
fields over finite fields,
- local uniformization (the local form of resolution of singularities.

Even if resolution of singularities in positive characteristic is proved 
soon (as several experts now believe), the model theoretic problem is not 
solved. We will need to know much more about the defect and the structure 
of valued function fields in several variables.

A question connected with both problems is: under which conditions is a 
ground field existentially closed in a function field? Is the existence of 
a rational place sufficient? It is if the ground field is perfect and 
large. We do not know whether the "perfect" condition can be dropped. I 
will discuss some background of this question.