We report on diode laser side-pumped, cw Nd:YAG rod lasers operating at pump powers up to 1 kW. With linear diode laser arrays as pump sources a pump power of 90 W/cm is realized. In multimode operation at 1064 nm, output powers of more than 300 W cw are observed. Applying a dynamically stable resonator design, an output power of more than 45 W in TEM00 mode operation is realized with an optical-to-optical efficiency of more than 11\%. Higher pump powers up to several 100 W/cm can be achieved by using fiber-coupled diode lasers as pump sources. Laser performance, thermal properties and possible applications of these laser systems will be discussed.