In this talk, we discuss two recent developments concerning sphere covering problems in high-dimensional Euclidean spaces. The first result improves the classical upper bound for lattice coverings by equal spheres: a new construction achieves density \(O(n\log^\beta n)\) with \(\beta\approx1.858\), improving Rogers' 1959 bound \(O(n\log^\alpha n)\) with \(\alpha\approx2.047\). The second result concerns random coverings by translates of convex bodies. For unit balls, it is shown that one can attain the best known asymptotic density \((1/2+o(1))\,n\ln n\) while simultaneously reducing the maximum overlap multiplicity to \(1.79556\,n\ln n\), improving a classical bound of Erd\H{o}s--Rogers. Furthermore, the work demonstrates intrinsic limitations of the standard random periodic method, which cannot achieve densities below \((1/2+o(1))\,n\ln n\) for any convex body, and identifies cubes as extremal cases with especially poor random covering performance.
