Sphere packings revisited

In this paper we survey most of the recent and often surprising results on packings of congruent spheres in d-dimensional spaces of constant curvature. The topics discussed are as follows: Hadwiger numbers of convex bodies and kissing numbers of spheres; Touching numbers of convex bodies; Newton numbers of convex bodies; One-sided Hadwiger and kissing numbers; Contact graphs of finite packings and the combinatorial Kepler problem; Isoperimetric problems for Voronoi cells, the strong dodecahedral conjecture and the truncated octahedral conjecture; The strong Kepler conjecture; Bounds on the density of sphere packings in higher dimensions; Solidity and uniform stability. Each topic is discussed in details along with some of the ”most wanted” research problems.