  For a given genus $g \geq 1$, we give lower bounds for the maximal number of
rational points on a smooth projective absolutely irreducible curve of genus
$g$ over ${\mathbb F}_q$. As a consequence of Katz-Sarnak theory, we first get
for any given $g>0$, any $\varepsilon>0$ and all $q$ large enough, the
existence of a curve of genus $g$ over ${\mathbb F}_q$ with at least $1+q+
(2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces
of Frobenius of hyperelliptic curves, we get a lower bound of the form
$1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit
constructions of towers of curves improve this result, with a bound of the form
$1+q+4 \sqrt{q} -32$ valid for all $g\ge 2$ and for all $q$.
