  We show that rings of $S$-integers of a global function field $K$ of odd
characteristic are first-order universally definable in $K$. This extends work
of Koenigsmann and Park who showed the same for $\mathbb{Z}$ in $\mathbb{Q}$
and the ring of integers in a number field, respectively.
  We also give another proof of a theorem of Poonen and show that the set of
non-squares in a global field of characteristic $\neq 2$ is diophantine.
Finally, we show that the set of pairs $(x,y)$ in $(K^{\times})^2$ such that
$x$ is not a norm in $K(\sqrt{y})$ is diophantine over $K$ for any global field
$K$ of characteristic $\neq 2$.
