The hydrogen dynamics in solid and in liquid LiBH4 was studied by means of incoherent quasielastic neutron scattering. Rotational jump diffusion of the BH4- subunits on the picosecond scale was observed in solid LiBH4. The characteristic time constant is significantly shortened when the system transforms from the low-temperature phase to the high-temperature phase at 383 K. In the molten phase of LiBH4 above 553 K, translational diffusion of the BH4- units is found. The measured diffusion coefficients are in the 10(-5)cm(2)/s range at temperatures around 700 K, which is in the same order of magnitude as the self-diffusion of liquid lithium or the diffusion of ions in molten alkali halides. The temperature dependence of the diffusion coefficient shows an Arrhenius behavior, with an activation energy of E-a = 88 meV and a prefactor of D-0 = 3.1 x 10(-4)cm(2)/s.