Pulsed frequency-doubled Nd3+ YAG laser beams were used to cut composites. The lasers are pre-prototypes of the diode-pumped lasers being developed and used in the Precision Laser Machining program. Cutting by conventional mechanical means is slow and can suffer from residual surface roughness and contamination. There is a strong interest in developing improved laser cutting processes having a speed exceeding 1 cm/s for composite materials, and having residual effects, such as roughness and widths of the melted and heat-affected zones, typically less than 25 mu m. The new test data presented here are produced by a nearly-diffraction limited 0.53 mu m beam with submicrosecond pulses and a typical repetition rate of 5 kHz. A kerf width of 100 mu m was produced by moving a f/6 focused beam spot at up to 5 cm/s with high pressure inert assist gas. The composite test samples varied from 0.023 up to 0.210 cm in thickness. The cold absorptances of the test samples varied from 0.16 to 0.94 at this wavelength, but increased considerably as the surface reached ablation temperature. Comparisons with earlier data obtained at 1.06 mu m wavelength on these same types of composite materials generally show an improvement ranging up to 5 times faster cutting speed for the 0.532 mu m green wavelength at the same average power, however, some materials had a slower cutting speed. An estimate of the correct Q* ablation parameter (J per g of material ablated) for each material was used to predict cutting performance. The ability to predict the cutting process from a simple ablation model coupled with the estimated Q* and the cold absorptance for each material is not completely successful. In some cases, and at cutting speeds over 1 cm/s, the finish quality almost met industrial requirements.