Ultrashort pulse lasers capable of delivering high energy pulses are limited in repetition rate. Using the combination of laser diodes to pump the laser and advanced cryogenic cooling we have demonstrated Joule-class picosecond lasers at record high repetition rate and average power.  Operation at 500 Hz has produced compressed 1 J pulses with a 5 ps pulse duration [1]. Preliminary tests of operation at 1 kHz have generated 1.4 kW of power in uncompressed pulses [2]. We have utilized this laser system to drive plasma-based soft x-ray lasers at a record 400 Hz, generating 0.15 mW of coherent light at 18.9 nm and 0.1 mW at 13.9 nm.