Ovals and Hyperovals in Desarguesian Nets

We determine the Desarguesian planes which hold $r$-nets with ovals and those which hold $r$-nets with hyperovals for every $r \le 7$.Comment: 21 pages, LaTeX. We have added a three-page section 6 (with non-existence results). We have also made explicit that ovals of Desarguesian nets are ``affinely equivalent'' under certain assumption