Metal halide perovskites (MHPs) have numerous advantages as light emitters such as high photoluminescence quantum efficiency with a direct bandgap, very narrow emission linewidth, high charge-carrier mobility, low energetic disorder, solution processability, simple color tuning, and low material cost. Based on these advantages, MHPs have recently shown unprecedented radical progress (maximum current efficiency from 0.3 to 42.9 cd A(-1)) in the field of light-emitting diodes. However, perovskite light-emitting diodes (PeLEDs) suffer from intrinsic instability of MHP materials and instability arising from the operation of the PeLEDs. Recently, many researchers have devoted efforts to overcome these instabilities. Here, the origins of the instability in PeLEDs are reviewed by categorizing it into two types: instability of (i) the MHP materials and (ii) the constituent layers and interfaces in PeLED devices. Then, the strategies to improve the stability of MHP materials and PeLEDs are critically reviewed, such as A-site cation engineering, Ruddlesden-Popper phase, suppression of ion migration with additives and blocking layers, fabrication of uniform bulk polycrystalline MHP layers, and fabrication of stable MHP nanoparticles. Based on this review of recent advances, future research directions and an outlook of PeLEDs for display applications are suggested.