By combining neutron inelastic scattering (NIS) and first-principles calculations, we have investigated the lattice dynamics of metal-organic framework-5 (MOF5). The structural stability of MOF5 was evaluated by calculating the three cubic elastic constants.We find that the shear modulus, c44=1.16 GPA, is unusually small, while two other moduli are relatively large (i.e., c11=29.42 GPa and c12=12.56 GPa). We predict that MOF5 is very close to structural instability and may yield interesting phases under high pressure and strain. The phonon dispersion curves and phonon density of states were directly calculated and our simulated NIS spectrum agrees very well with our experimental data. Several interesting phonon modes are discussed, including the softest twisting modes of the organic linker.