The continuous miniaturization and integration of pixelated devices have become a main trend in the field of display. Micro light-emitting diode (micro-LED) display is composed of an array of LEDs that are sub-50-micrometers in length. It has huge advantages in brightness, resolution, contrast, power consumption, lifetime, response speed and reliability compared with liquid crystal display (LCD) and organic LED (OLED) display. Consequently, micro-LED display is regarded as the next-generation display technology with high potential applications, such as virtual reality (VR), augmented reality (AR), mobile phones, tablet computers, high-definition TVs and wearable devices. Currently, the combination of commercial 5G communication technology with VR/AR display, ultra high definition video technologies will further prompt the development of micro-LED display industry. However, some basic scientific and technological problems in micro-LED display remain to be resolved. As the chip size shrinks to below 50 μm, some problems that are not serious for large-sized LEDs appear for micro-LEDs. These problems include crystalline defects, wavelength uniformity, full-color emmision, massively tranferring and testing, etc. In the past two decades, various solutions to those problems have been proposed, which have greatly promoted the progress of micro-LED display. In this paper, an overview of micro-LED display since 2000 is given firstly, which includes the main research results and application achievements. Secondly the issues involved in the wafer epitaxy and chip process of micro-LEDs and possible solutions are discussed based on the display application in detail. The surface state induced by the dangling bonds and dry etching damages are concerned for the nonradiative recombination at a low injection level. The remedies are provided for those surface states, such as atomic-layer deposition and neutral beam etching. Some methods to reduce the threading dislocation and suppress the polarization field are summarized for micro-LED epitaxial growth. Moreover, the GaN-based LEDs on Si (100) substrate are also introduced for the future integration of micro-LEDs into the Si-based integrated circuits. As to the wavelength uniformity, the MOCVD equipment and growth technology including the laser treatment are discussed. In the chip processing part, the full-color display, mass transfer and effective inspection technology are discussed. Assembling RGB individual LEDs, quantum dot phosphor material and nanocoloumn LEDs are different routes for full-color display. Their trends in the future are provided. The pick and place, laser lift-off technologies, are strengthened in the massively transferring for micro-LEDs. In the massively and rapidly inspection technologies, the photoluminscence combined with Raman scattering, the electroluminescence combined with digital camera are discussed. Finally, the summary and outlook in these issues are also provided.