Cubic zirconia containing 10 mol% yttria was deformed along the hard (100) direction and materials with 15 and 20 mol% yttria along the soft 112 orientation between 1400 ^°C and the lowest possible temperatures without brittle failure. The 10% material shows along (100) a strong increase in the flow stress at decreasing temperature, while the materials with 15 and 20 mol% yttria along (112) exhibit a higher flow stress with a weak temperature dependence. The activation parameters of deformation were measured by stress relaxation and temperature change tests. The microstructures are studied by high-voltage transmission electron microscopy. The results are interpreted by long-range interactions between the dislocations, the Peierls mechanism at the lowest temperatures, precipitation hardening between 500 ^°C and 800 ^°C, and by recovery-controlled deformation at high temperatures. The occurrence of plastic instabilities are described in Part II [phys. stat. sol. (a) 201, No. 1 (2004)] of this paper.