  The nuclear and magnetic structure and full magnon dispersions of yttrium
iron garnet Y$_3$Fe$_5$O$_{12}$ have been studied by neutron scattering. The
refined nuclear structure is distorted to a trigonal space group of $R\bar{3}$.
The highest-energy dispersion extends up to 86 meV. The observed dispersions
are reproduced by a simple model with three nearest-neighbor-exchange integrals
between 16$a$ (octahedral) and 24$d$ (tetrahedral) sites, $J_{aa}$, $J_{ad}$,
and $J_{dd}$, which are estimated to be 0.00$\pm$0.05, $-$2.90$\pm$0.07, and
$-$0.35$\pm$0.08 meV, respectively. The lowest-energy dispersion below 14 meV
exhibits a quadratic dispersion as expected from ferromagnetic magnons. The
imaginary part of $q$-integrated dynamical spin susceptibility $\chi$"($E$)
exhibits a square-root energy-dependence in the low energies. The magnon
density of state is estimated from the $\chi$"($E$) obtained on an absolute
scale. The value is consistent with a single polarization mode for the magnon
branch expected theoretically.
