  We study arithmetical and geometrical properties of {\it maximal curves},
that is, curves defined over the finite field $\mathbb F_{q^2}$ whose number of
$\mathbb F_{q^2}$-rational points reachs the Hasse-Weil upper bound. Under a
hypothesis on non-gaps at rational points we prove that maximal curves are
$\mathbb F_{q^2}$-isomorphic to $y^q+y=x^m$ for some $m\in \mathbb Z^+$.
