The author gives an overview concerning upper bounds for the numbers of rational points on a curve of genus g, over a finite field of q elements. In general one has the bound given by Weil’s theorem. There are more precise results of the following nature: (i) asymptotic results: Fix q, and let \(g\to \infty\); (ii) results for \(q=2\); (iii) results for \(g=1,2,3\).