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We study peculiarities of phase transitions in zirconium and properties of the high-temperature β-Zr phase. To get a more detailed understanding of the structure and thermodynamic characteristics of zirconium, we perform atomistic simulations with two different interatomic potentials. Both potentials demonstrate an unstable behavior of β-Zr phase at low temperatures but explain this phenomenon by substantially different reasons. For one of the potentials, the mechanical instability takes place, and for the other potential the instability of β-Zr is purely dynamic. Review of the available experimental data shows that it is more correct to describe β-Zr through the low-temperature dynamic instability. The structure peculiarity discussed for β-Zr leads to a local non-cubic symmetry of this phase and low formation energy of the self-interstitial atoms. The latter leads to fast atomic self-diffusion that is consistent with existing data. We also perform deformation tests for the atomistic models of β-Zr-Nb alloys taking into account the studied details of α-β transition.