Organic-inorganic hybrid perovskite materials are receiving increasing attention and becoming star materials on account of their unique and intriguing optical and electrical properties, such as high molar extinction coefficient, wide absorption spectrum, low excitonic binding energy, ambipolar carrier transport property, long carrier diffusion length, and high defects tolerance. Although a high power conversion efficiency (PCE) of up to 22.7% is certified for perovskite solar cells (PSCs), it is still far from the theoretical Shockley-Queisser limit efficiency (30.5%). Obviously, trap-assisted nonradiative (also called Shockley-Read-Hall, SRH) recombination in perovskite films and interface recombination should be mainly responsible for the above efficiency distance. Here, recent research advancements in suppressing bulk SRH recombination and interface recombination are systematically investigated. For reducing SRH recombination in the films, engineering perovskite composition, additives, dimensionality, grain orientation, nonstoichiometric approach, precursor solution, and post-treatment are explored. The focus herein is on the recombination at perovskite/electron-transporting material and perovskite/hole-transporting material interfaces in normal or inverted PSCs. Strategies for suppressing bulk and interface recombination are described. Additionally, the effect of trap-assisted nonradiative recombination on hysteresis and stability of PSCs is discussed. Finally, possible solutions and reasonable prospects for suppressing recombination losses are presented.