  Some new results on plane F_{q^2}-maximal curves are stated and proved. It is
known that the degree d of such curves is upper bounded by q+1 and that d=q+1
if and only if the curve is F_{q^2}-isomorphic to the Hermitian. We show that
d\le q+1 can be improved to d\le (q+2)/2 apart from the case d=q+1 or q\le 5.
This upper bound turns out to be sharp for q odd. We also study the maximality
of Hurwitz curves of degree n+1. We show that they are F_{q^2}-maximal if and
only if (q+1) divides (n^2-n+1). Such a criterion is extended to a wider family
of curves.
