Modeling of static and flowing-gas subsonic, transonic and supersonic Cs and K Ti:Sapphire and diode pumped alkali lasers (DPALs) is reported. A simple optical model applied to the static K and Cs lasers shows good agreement between the calculated and measured dependence of the laser power on the incident pump power. The model reproduces the observed threshold pump power in K DPAL which is much higher than that predicted by standard models of the DPAL. Scaling up flowing-gas DPALs to megawatt class power is studied using accurate three-dimensional computational fluid dynamics model, taking into account the effects of temperature rise and losses of alkali atoms due to ionization. Both the maximum achievable power and laser beam quality are estimated for Cs and K lasers. The performance of subsonic and, in particular, supersonic DPALs is compared with that of transonic, where supersonic nozzle and diffuser are spared and high power mechanical pump (needed for recovery of the gas total pressure which strongly drops in the diffuser), is not required for continuous closed cycle operation. For pumping by beams of the same rectangular cross section, comparison between end-pumping and transverse-pumping shows that the output power is not affected by the pump geometry, however, the intensity of the output laser beam in the case of transverse-pumped DPALs is strongly non-uniform in the laser beam cross section resulting in higher brightness and better beam quality in the far field for the end-pumping geometry where the intensity of the output beam is uniform.