Author’s abstract: We review some recent progress in the packing of convex bodies in the d-dimensional Euclidean space. After some historical perspective on low dimensional packing, the first and major part is an introduction to the decisive breakthrough by M. Viazovska settling the densest (hyper)sphere packings for \(d=8\) and 24. Before outlining her proof, the relevant ground works by others for sphere packings in higher dimension as well as minimal background on modular forms is presented. We then proceed to some samples of rigorous and non-rigorous results in the asymptotics (in \(d\)) of the maximal packing density of spheres. At the end we discuss advances in the packings of convex polyhedra for \(d=3\) and in particular the resolution of Ulam’s Conjecture. Along the way some closely related results on kissing numbers are indicated.