We discuss progress towards a kilowatt class CW Yb:YAG cryogenic laser. Cryogenically-cooled crystalline solid-state lasers, and Yb:YAG lasers in particular, are attractive sources of scalable CW output power with very high wallplug efficiency and excellent beam-quality that is independent of the output power. Results are presented for a high power Yb:YAG oscillator that has produced over 550 W of output power with good slope and optical-optical efficiencies while maintaining single transverse mode output. We also describe a new oscillator-amplifier cryogenic Yb:YAG system nearing completion, that will build on the work presented here and result in CW power output of > 1 kW while maintaining near-diffraction-limited beam quality. The oscillator described here consists of a distributed array of seven highly-doped thin Yb:YAG-sapphire disks in a folded multiple-Z resonator. Individual disks are pumped from opposite sides using 100 W fiber-coupled 940 nm pump diodes. The laser system produces a near-diffraction-limited TEM 00 output beam with the aid of an active conduction-cooling design. In addition, the device can be scaled to very high average power in an oscillator-amplifier configuration, by increasing the number and diameter of the thin disks, and by increasing the power of the pump diodes with only minor modifications to the current design. We will present experimental results including output power, threshold power, and slope and optical-optical efficiencies.