  This review describes the diversity of jammed configurations attainable by
frictionless convex nonoverlapping (hard) particles in Euclidean spaces and for
that purpose it stresses individual-packing geometric analysis. A fundamental
feature of that diversity is the necessity to classify individual jammed
configurations according to whether they are locally, collectively, or strictly
jammed. Each of these categories contains a multitude of jammed configurations
spanning a wide and (in the large system limit) continuous range of intensive
properties, including packing fraction $\phi$, mean contact number $Z$, and
several scalar order metrics. Application of these analytical tools to spheres
in three dimensions (an analog to the venerable Ising model) covers a myriad of
jammed states, including maximally dense packings (as Kepler conjectured),
low-density strictly-jammed tunneled crystals, and a substantial family of
amorphous packings. With respect to the last of these, the current approach
displaces the historically prominent but ambiguous idea of ``random close
packing" (RCP) with the precise concept of ``maximally random jamming" (MRJ).
This review also covers recent advances in understanding jammed packings of
polydisperse sphere mixtures, as well as convex nonspherical particles, e.g.,
ellipsoids, ``superballs", and polyhedra. Because of their relevance to
error-correcting codes and information theory, sphere packings in
high-dimensional Euclidean spaces have been included as well. We also make some
remarks about packings in (curved) non-Euclidean spaces. In closing this
review, several basic open questions for future research to consider have been
identified.
