We discuss two problems about lattice points. (1) Motivated by problems in mathematical physics, Antunes Freitas proved in 2012 that among all axis-parallel ellipses in the plane centered at the origin and having fixed area, the one containing the most lattice points asymptotically converges to a circle as area becomes large. We give a far-ranging generalization to convex bodies with nonvanishing Gauss curvature. The methods are based on Fourier analysis. (2) Secondly, in joint work with S. Steinerberger, we answer a question posed by Laugesen Liu about right-angled triangles in the plane capturing many lattice points. Instead of a single optimal shape, there is a countably infinite limit set of triangles each of which capture a maximal number of positive lattice points for arbitrarily large areas. Moreover, this limit set is fractal with Minkowski dimension at most 3/4. The proof involves elements from combinatorics and dynamical systems.