I demonstrate why the venerable 50-year old notion of "random close packing" (RCP) of spheres (the putative "random" analog of Kepler's conjecture) is mathematically ill-defined. To replace this traditional notion, we introduce a new concept of a maximally random jammed (MRJ) state, which can be made precise. The latter idea rests on devising precise meanings for "jamming" and "randomness," which I describe. We show that there are random packings of ellipsoids in three-dimensional Euclidean space that closely approach the densest lattice packing. Moreover, we have discovered a periodic (non-lattice) ellipsoid packing with the highest known density and one in which the ellipsoids are not highly degenerate in shape (i.e., highly eccentric). Present results do not exclude the possibility that even denser periodic packings of ellipsoids could be found, and that a corresponding Kepler-like conjecture could be formulated for ellipsoids.
Finally, I discuss a new approach to derive lower bounds on the density of random sphere packings in d-dimensional Euclidean space. One of these bounds is sharper than the well-known (nonconstructive) Minkowski lower bound that applies to the densest lattice packings in d-dimensional Euclidean space. Another improved bound is the sharpest to date.