Generalising classical questions about regular polygons with vertices on a plane lattice, we are interested in pairs of points A,B on a lattice such that the angle \(\hat{AOB}\) is a rational multiple of \(\pi\). This problem leads to diophantine-trigonometric equations that in turn involve the study of rational points on curves of genus 0,1,2,3,5. I will present the full classification of plane lattices according to how many independent rational angles they contain and in which configurations they appear. This is a joint work with R. Dvornicich, D. Lombardo and U. Zannier