Abstract: Undefinedindent How many points with integral coordinates are there in the disk with centre $O$ and radius $R$, or, in other wordes, what is the number of integral solutions $(x, y)$ to the inequation $x^2+y^2undefinedle R$? It is not difficult to see that this number of points, say $N(R)$, is closer and closer to $undefinedpi R^2$ (the area of the disk), as $R$ becomes larger and larger. How precise is this relation, what can we say about the quantity $|N(R) – undefinedpi R^2|$? We shall discuss this question, as well as generalizations of it, including the surprising result obtained by a high-school teacher 50 years ago undefined

undefinedindent We shall also consider the number of integral points on a curve $undefinedmathcal{C}$ in the euclidean plane. It is here natural to compare that number of points to the length of the curve (a good bound for rational straight lines) but this number drastically drops as soon as $undefinedmathcal{C}$ is simply assumed to be strictly convex. We shall describe some very recent contributions to this problem, obtained by using only elementary arguments (joint work of Grekos, Urbis and the lecturer).