Gravitational Waves (GWs) are ripples of space-time that propagate across the universe at the speed of light. GWs detection is one of the most important frontiers of physics today. Ground-based laser interferometer gravitational wave detectors, such as LIGO, Virgo and KAGRA, have successfully confirmed the existence of GWs. GWs have become a new window for a human to observe the universe. Due to the influence of ground vibration noise, terrestrial gravity gradient noise and other factors, ground-based detectors are not sensitive to GWs below 1 Hz. Space-based detectors are free from such noises and can be made very large, thereby expanding the frequency range downwards to 10^-4 Hz, where exciting GW sources are waiting to be explored. Space-based GW detectors are expected to bring even more information about the universe through low-frequency GWs. A highly stable and long-lifetime laser system is a key component of the space-based GW detector, and the output power, intensity noise, frequency noise and other properties of the laser directly affect the sensitivity of the space-based GW detector. Thus, space-based GW detectors put forward much higher and stricter requirements for the laser. After widely experimental and industrial surveys, the baseline architecture for the laser for space-based GW detector consists of a low-power, low-noise master oscillator followed by a power amplifier with 2~10 W continuous-wave output. So, lasers consisting of a Master Oscillator and a Power Amplifier (MOPA) have been identified as the most-promising architecture for the space-based GW detector. In this review, we focused on the research progress of low-noise MOPA lasers in LISA, TianQin and Taiji missions. As in other applications of precision interferometry, 1064 nm was chosen as the laser wavelength, due to the availability of high-quality bulk optics and the traditional low-noise Nd:YAG laser source represented by the Non-Planar Ring Oscillator (NPRO).The performance of different MOPA lasers for LISA injected by NPRO(m-NPRO), fiber laser and External Cavity Diode Laser (ECDL) were compared, and the m-NPRO has been identified as the most-promising MO architecture for the LISA laser, and the baseline architecture consists of a low-power, low-noise m-NPRO followed by a diode-pumped Yb-fiber amplifier with ~2 W output. As for the Relative Intensity Noise (RIN), the m-NPRO meets the LISA requirement. As to the Chinese space-based GW detection projects, the research progress and spatial test results of lasers were elaborated. A DBR laser was designed for the TianQin-1 mission as the space laser in the laser interferometer, and was successfully verified in the TianQin-1 mission. The laser passed the environment performance verification under the aerospace standard. Meanwhile, a high stability laser source at 1064 nm for Taiji-1 satellite was reported. The key component of the laser source was a NonPlanar Ring Oscillator (NPRO) solid laser with linewidth of 260 Hz. The frequency noise and power noise of Laser source were greatly improved by applying precision driving current control and temperature control. Furthermore, around the requirements of the noise performance of lasers by space-based GW detectors, the principle and progress of intensity noise suppression and frequency noise suppression were described, and the main achievements of photoelectric feedback suppression technology for intensity noise suppression and PDH technology for frequency noise suppression were discussed respectively. We designed a laser system with the characteristics of narrow-linewidth, low-noise and polarization-maintaining. In our experiment setup, optoelectronic feedback suppression technology was used to suppress pump LD fluctuations using low-noise photodetectors and PID controller. In the frequency domain, the relative intensity noise is reduced at the level of 10^-3 Hz^-1/2@1 mHz. Finally, the research on low noise lasers for space-based GW detection in China is prospected, and the development direction of low noise lasers is proposed.