  A hyperoval in the projective plane $\mathbb{P}^2(\mathbb{F}_q)$ is a set of
$q+2$ points no three of which are collinear. Hyperovals have been studied
extensively since the 1950s with the ultimate goal of establishing a complete
classification. It is well known that hyperovals in
$\mathbb{P}^2(\mathbb{F}_q)$ are in one-to-one correspondence to polynomials
with certain properties, called o-polynomials of $\mathbb{F}_q$. We classify
o-polynomials of $\mathbb{F}_q$ of degree less than $\frac12q^{1/4}$. As a
corollary we obtain a complete classification of exceptional o-polynomials,
namely polynomials over $\mathbb{F}_q$ that are o-polynomials of infinitely
many extensions of $\mathbb{F}_q$.
