Atmospheric predictability research is the basis for weather and climate prediction. Under the background of global warming, meso/micro-scale extreme weather such as heavy rain and severe convection occur more frequently in recent years, and its predictability has attracted widely attention. After a brief review of the history of atmospheric predictability research, this paper systematically reviews and summarizes the latest advances on the predictability of heavy rain and strong convection over the last 20 years. The main research methods for meso/micro-scale predictability and their differences with traditional large-scale weather predictability research are first discussed. Then, the primary initial error growth mechanism (error upscaling under deep moist convection) has been elaborated in detail and some arguments (error downscaling, error upscaling and downscaling coexisting) are discussed. The influences of errors in NWP models and convective environment to the practical predictability, as well as some recent mesoscale predictability experiments are also highlighted. Finally, this paper briefly discusses the current problems, challenges and future directions on predictability research of heavy rain and severe convection.