The experimental research regarding grooving and cutting the sapphire substrate by nanosecond laser with wavelength of 355 nm and picosecond laser with wavelength of 1064 nm is presented. The influence of adjusted processing parameters on the depth, width and quality of grooves via controlling the variable method is studied. Furthermore, the comparative analysis shows that the thermal effect of 355 nm nanosecond laser processing is very big, and the etching efficiency is low, which cannot achieve graphs cutting of sapphire substrate even if its peak power density reaches 109 W/cm2. However, 1064 nm picosecond infrared laser cutting quality of micro groove is very high, which can realize completely “cold” processing. Eventually, the results show that the graphs cutting on sapphire substrate can be realized with the laser peak power density of 1012 W/cm2, The high-quality graphs cutting on the blank of sapphire substrate with 0.55 mm thickness is realized by 1064 nm infrared laser with average power 1 W, single pulse energy 1 mJ, and pulse width 15 ps.