Using cryogenic laser technology, it is now possible to design and demonstrate lasers that have concomitant high average and peak powers, with near-diffraction-limited beam quality. We refer to these new laser systems as HAPP lasers. In this paper, we review important laser crystal materials properties at cryogenic temperature, with an emphasis on Yb lasers, and discuss the important design considerations, including the laser-induced damage threshold, nonlinear effects and thermal effects. A comprehensive model is presented to describe diode pulsed pumping with arbitrary duration and repetition rate, and is used with the Frantz–Nodvik equation to describe, to first order, the performance of HAPP laser systems. A computer code with representative results is also described.