We study the charge transport mechanism in ferroelectric Hf0.5Zr0.5O2 thin films. Transport properties of Hf0.5Zr0.5O2 are described by phonon-assisted tunnelling between traps. Comparison with transport properties of amorphous Hf0.5Zr0.5O2 demonstrates that the transport mechanism does not depend on the structure. The thermal and optical trap energies 1.25 eV and 2.5 eV, respectively, in Hf0.5Zr0.5O2 were determined based on comparison of experimentally measured data on transport with simulations within phonon-assisted tunnelling between traps. We found that the trap density in ferroelectric Hf0.5Zr0.5O2 is slightly less than one in amorphous Hf0.5Zr0.5O2. A hypothesis that oxygen vacancies are responsible for the charge transport in Hf0.5Zr0.5O2 is confirmed by ab initio simulation of electronic structure.