Picosecond laser with high-repetition-rate and high pulse energy is widely favorite in many scientific and industrial applications. Some nonlinear crystals can be used to efficiently convert a near-infrared laser into a green laser or an ultraviolet laser which has a higher photon energy and a smaller focal area. Especially for high-quality and high-speed transparent hard material fabrication, green or ultraviolet picosecond laser has been found to possess unique advantages. In this paper, the experiments on high-efficiency second-harmonic-generation (SHG) and third-harmonic-generation (THG) by using a home-made all-solid-state picosecond laser amplifier and an LBO crystal are reported. The picosecond laser amplifier consists of a seed source, a regenerative amplifier and a two-stage single-pass amplifier. The seed source is a commercial all-solid-state picosecond oscillator with a pulse duration of 8.3 ps and a repetition rate of 68 MHz. The repetition rate is reduced from 68 MHz to 500 kHz by an electro-optic Pockels cell (PC), and the period doubling bifurcation is minimized by reducing the duration of high voltage in PC. Both the regenerative amplifier and the two-stage single-pass amplifier are pumped by three 30-W continuous-wave fiber-coupled laser diodes. After the regenerative amplifier, the seed laser is amplified to 4.86 W with a repetition rate of 500 kHz at 1064 nm. Then the laser power is increased to 23.2 W by a two-stage single-pass amplifier, and the M2 value of the amplified laser in the X direction and in the Y direction are 1.330 and 1.235, respectively. The final pulse duration is 13.4 ps, which is slightly stretched in the amplification chain compared with the seed pulse duration (8.3 ps). For high-efficiency SHG and THG from near-infrared to green and ultraviolet, we carefully study the optical characteristics of some nonlinear crystals, such as LBO, BBO, BIBO, CLBO, etc., and we find that the LBO crystal, which has a high damage threshold, small walk-off and high nonlinear coefficient, is the best choice for both SHG and THG. Then the parameters of the two crystals for SHG and THG are specially designed according to the phase matching condition, the walk-off and the laser parameter. As a result, a 4-mm-long type-I phase matching LBO with cutting angle of θ = 90° and φ = 11.6° is used for SHG, and a 3-mm-long type-II phase matching LBO with cutting angle of θ = 42.2° and φ = 90° is used for THG. Finally, we realize high-efficiency frequency conversion with SHG power of 12.7 W at 532 nm and THG power of 9.25 W at 355 nm. The corresponding optical-optical conversion efficiencies reach 54.7% and 39.6%, respectively.