  We prove that the existential theory of any function field $K$ of
characteristic $p> 0$ is undecidable in the language of rings provided that the
constant field does not contain the algebraic closure of a finite field. We
also extend the undecidability proof for function fields of higher
transcendence degree to characteristic 2 and show that the first-order theory
of {\bf any} function field of positive characteristic is undecidable in the
language of rings without parameters.
