The accuracy of approximation of the distribution of sums of independent random vectors by Gaussian distributions is fundamental for many applications. In particular approximations of the distribution of norms of such sums are of interest. It is shown how progress in this question is closely linked to classical problems in the geometry of numbers for balls and ellipsoids by Landau and Hardy. Recent optimal results in lattice point problems and the equidistribution on orbits of unipotent subgroups have led to progress in this problem for dimension 5 and larger.