  Particle packing problems have fascinated people since the dawn of
civilization, and continue to intrigue mathematicians and scientists. Resurgent
interest has been spurred by the recent proof of Kepler's conjecture: the
face-centered cubic lattice provides the densest packing of equal spheres with
a packing fraction $\phi\approx0.7405$ \cite{Kepler_Hales}. Here we report on
the densest known packings of congruent ellipsoids. The family of new packings
are crystal (periodic) arrangements of nearly spherically-shaped ellipsoids,
and always surpass the densest lattice packing. A remarkable maximum density of
$\phi\approx0.7707$ is achieved for both prolate and oblate ellipsoids with
aspect ratios of $\sqrt{3}$ and $1/\sqrt{3}$, respectively, and each ellipsoid
has 14 touching neighbors. Present results do not exclude the possibility that
even denser crystal packings of ellipsoids could be found, and that a
corresponding Kepler-like conjecture could be formulated for ellipsoids.
