The authors study valued function fields, i.e. function fields of one variable with valuation on the constant field and a prolongation to the function fields such that the corresponding residue fields are again function fields in one variable. They introduce the vector space defect for a non-archimedean valued vector space over a valued field (which generalizes classical invariants of the valued field extensions). This vector space defect can be approximated by quotients of the degree of large positive divisors by the degrees of certain residual divisors. As an application they show that equality of the genus and residual genus, greater than 1, characterizes the valued function fields having good reduction. Finally, they show that the vector space defect is closely related to the henselian defect of the valued function field and show that over a henselian base field these two defects coincide. In the last section the authors prove the inequality \[ \chi (F| K)\leq 1-s+\sum_{1\leq i\leq s}e_ i\delta_ i\chi (Fv_ i| Kv_ i) \] where \((F| K,v_ i)\) are valued function fields with \(v_ i| K=v_ j| K=v\) on the exact constant field K and \(Fv_ i\) and \(Kv_ i\) are corresponding residue fields.