One of the basic problems in discrete geometry is to determine the most
efficient packing of congruent replicas of a given convex set $K$ in the plane
or in space. The most commonly used measure of efficiency is density. Several
types of the problem arise depending on the type of isometries allowed for the
packing: packing by translates, lattice packing, translates and point
reflections, or all isometries. Due to its connections with number theory,
crystallography, etc., lattice packing has been studied most extensively. In
two dimensions the theory is fairly well developed, and there are several
significant results on lattice packing in three dimensions as well. This
article surveys the known results, focusing on the most recent progress. Also,
many new problems are stated, indicating directions in which future development
of the general packing theory in three dimensions seems feasible.