Comets are most pristine materials in the solar system, which were formed by accumulation of icy dusts from the solar nebula or molecular clouds. Comets mainly consist of water ice, silicates, and organic refractory materials, and also contain abundant organic molecules. The astronomical observation revealed that more than 27 different organic molecules exist in comets. It also revealed the isotopic compositions of some organic molecules. In addition, in situ analyses of comets by cometary explorers and the analyses of returned cometary samples have provided further information about cometary organic molecules. These findings showed that comets contain important biomolecules and their precursors such as amino acid, amine, and hydrogen cyanide. On the other hand, the new knowledge raised new questions about the distinct isotopic compositions of comets. In this review, I summarize the latest knowledge about cometary organic molecules from astronomical observation and two recent successful space missions: Stardust mission and Rosetta mission. Then, I discuss the role of comets in molecular evolution of organic molecules that lead to the origins of life on the early Earth, with special emphasis on the effect of impact shock on organic molecules at comet impacts.