Our recent progress in the development of high energy / high average power, chirped pulse amplification laser systems based on diode-pumped, cryogenically-cooled Yb:YAG amplifiers is discussed, including the demonstration of a laser that produces 1 Joule, sub-10 picosecond duration, λ = 1.03μm pulses at 500 Hz repetition rate. This compact, all-diodepumped laser combines a mode-locked Yb:KYW oscillator and a water-cooled Yb:YAG preamplifer with two cryogenic power amplification stages to produce 1.5 Joule pulses with high beam quality which are subsequently compressed. This laser system occupies an optical table area of less than 1.5x3m2. This laser was employed to pump plasma-based soft x-ray lasers at λ = 10-20nm at repetition rates ≥100 Hz. To accomplish this, temporally-shaped pulses were focused at grazing incidence into a high aspect ratio line focus using cylindrical optics on a high shot capacity rotating metal target. This results in an elongated plasma amplifier that produces microjoule pulses at several narrow-linewidth EUV wavelengths between λ = 109Å and 189Å. The resulting fraction of a milliwatt average powers are the highest reported to date for a compact, coherent source operating at these wavelengths, to the best of our knowledge.