Molecular clouds (MCs) in space are the birthplace of various molecular species. Chemical reactions occurring on the cryogenic surfaces of cosmic icy dust grains have been considered to play important roles in the formation of these species. Radical reactions are crucial because they often have low barriers and thus proceed even at low temperatures such as ∼10 K. Since the 2000s, laboratory experiments conducted under low-temperature, high-vacuum conditions that mimic MC environments have revealed the elementary physicochemical processes on icy dust grains. In this review, experiments conducted by our group in this context are explored, with a focus on radical reactions on the surface of icy dust analogues, leading to the formation of astronomically abundant molecules such as H2, H2O, H2CO, and CH3OH and deuterium fractionation processes. The development of highly sensitive, non-destructive methods for detecting adsorbates and their utilization for clarifying the behavior of free radicals on ice, which contribute to the formation of complex organic molecules, are also described.