The high trap density (generally 1016-1018 cm-3) in organic solar cells (OSCs) brings about the localization of charge carriers and reduced charge carrier lifetime, mainly due to the weak intermolecular interactions of organic semiconductors resulting in their relatively poor crystallinity, which leads to low charge carrier mobilities and intense non-radiative recombination, thus impeding the further improvement of power conversion efficiencies (PCEs). Therefore, trap suppression is crucial to boost the performance of OSCs, and improving the crystallinity of donor/acceptor materials and enhancing the molecular order in devices can contribute to the trap suppression in OSCs. In this feature article, we summarize the recent advances of trap suppression in OSCs by material design and device engineering, and further outline possible development directions for trap suppression to enhance PCEs of OSCs.