For a given genus g ≥ 1, we give lower bounds for the maximal number of rational pointson a smooth projective absolutely irreducible curve of genus g over F_q. As a consequenceof Katz–Sarnak theory, we first get for any given g > 0, any ε > 0 and all q large enough,the existence of a curve of genus g over F_q with at least 1 + q + (2g − ε)√q rational points.Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lowerbound of the form 1 + q + 1.71√q valid for g ≥ 3 and odd q ≥ 11. Finally, explicit constructions of towers of curves improve this result: We show that the bound 1 + q + 4√q − 32 isvalid for all g ≥ 2 and for all q.