  Let $f$ be an $\mathbb{F}_q$-linear function over $\mathbb{F}_{q^n}$. If the
$\mathbb{F}_q$-subspace $U= \{ (x^{q^t}, f(x)) : x\in \mathbb{F}_{q^n} \}$
defines a maximum scattered linear set, then we call $f$ a scattered polynomial
of index $t$. As these polynomials appear to be very rare, it is natural to
look for some classification of them. We say a function $f$ is an exceptional
scattered polynomial of index $t$ if the subspace $U$ associated with $f$
defines a maximum scattered linear set in $\mathrm{PG}(1, q^{mn})$ for
infinitely many $m$. Our main results are the complete classifications of
exceptional scattered monic polynomials of index $0$ (for $q>5$) and of index
$1$. The strategy applied here is to convert the original question into a
special type of algebraic curves and then to use the intersection theory and
the Hasse-Weil theorem to derive contradictions.
