  The present work surveys problems in $n$-dimensional space with $n$ large.
Early development in the study of packing and covering in high dimensions was
motivated by the geometry of numbers. Subsequent results, such as the discovery
of the Leech lattice and the linear programming bound, which culminated in the
recent solution of the sphere packing problem in dimensions 8 and 24, were
influenced by coding theory. After mentioning the known results concerning
existence of economical packings and coverings we discuss the different methods
yielding upper bounds for the density of packing congruent balls. We summarize
the few results on upper bounds for the packing density of general convex
bodies. The paper closes with some remarks on the structure of optimal
arrangements.
