Ytterbium (Yb)-ions-doped sesquioxide crystal is an attractive gain medium for a tunable and pulsed laser owing to its high thermal conductivity. In particular, it has been identified that Yb:LuScO3 has the largest energy storage property compared with other sesquioxide crystals, which is favorable for passive Q switching. In this Letter, continuous wave (CW) and the first, to the best of our knowledge, passively Q switched laser operations were demonstrated with a Yb:LuScO3 crystal. For CW laser operation, it generated the maximum output power of 8.68 W, corresponding to a slope efficiency up to 78.3%. Using Cr:YAG crystals as saturable absorbers, stable passive Q switching lasers were obtained with the Yb:LuScO3 crystal. Both the CW and Q switched lasers operate on the strongest fluorescence emission peak of 1038 nm. With Cr:YAG as the saturable absorber, efficient passively Q switched lasers with a slope efficiency of 45% were obtained with the pulse width, pulse energy, and peak power of 5.9 ns, 116 µJ, and 18.5 kW, respectively.