(An official version and a preprint version are available online.)

We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant c_q with the following property: for every non-negative integer g, there is a genus-g curve over F_q with at least c_q g rational points over F_q. Moreover, we show that there exists a positive constant d such that for every q we can choose c_q=d log q. We show also that there is a positive constant c such that for every q and every positive integer n, and for every sufficiently large g, there is a genus-g curve over F_q that has at least cg/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (Z/nZ)^r for some r>cg/n.