The metastable orthorhombic phase of hafnia is generally obtained in polycrystalline films, whereas in epitaxial films its formation has been much less investigated. We have grown Hf0.5Zr0.5O2 films by pulsed laser deposition and the growth window (temperature and oxygen pressure during deposition, and film thickness) for epitaxial stabilization of the ferroelectric phase is mapped. The remnant ferroelectric polarization, up to around 24 μC/cm2, depends on the amount of orthorhombic phase and strain, and increases with temperature and pressure for fixed film thickness. Leakage current decreases by increasing thickness and temperature, and particularly by reducing oxygen pressure. The coercive electric field (EC) depends on thickness (t) according the EC - t-2/3 scaling, which is observed by the first time in ferroelectric hafnia, and the scaling extends to thickness below 5 nm. The proven ability to tailor functional properties of high quality epitaxial ferroelectric Hf0.5Zr0.5O2 films paves the way towards progress in understanding their ferroelectric properties and prototyping devices.