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 $Rbar{3}$. 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.
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