Advances in experimental techniques, especially the development of the CRESU (Cinétique de Réaction en Ecoulement Supersonique Uniforme) method, allow many gas-phase molecular processes to be studied at very low temperatures. This Review focuses on the reactions of molecular and atomic radicals with neutral molecules. Rate constants for almost 50 such reactions have been measured at temperatures as low as 13 K by using the CRESU method. The surprising demonstration that so many reactions between electrically neutral species can be extremely rapid at these very low temperatures has excited interest both from theoreticians and from those seeking to understand the chemistry that gives rise to the 135 or so molecules that are present in low-temperature molecular clouds in the interstellar medium. Theoretical treatments of these reactions are based on the idea that a reaction occurs when the long-range potential between the reagent species brings them into close contact. The astrochemical context, theoretical studies, and the determination of the rate constants of these low-temperature reactions are critically discussed.