The interdisciplinary science of astrochemistry is 45 years of age, if we pinpoint its origin to have occurred when the first polyatomic molecules were detected in the interstellar gas. Since that time, the field has grown remarkably from an esoteric area of research to one that unites scientists around the globe. Almost 200 different molecules have been detected in the gas-phase of interstellar clouds, mainly by rotational spectroscopy, while dust particles and their icy mantles in colder regions can be probed by vibrational spectroscopy. Astrochemistry is exciting to scientists in a number of different fields. Astronomers are interested in molecular spectra from the heavens because such spectra are excellent probes of the physical conditions where molecules exist, while chemists are interested in the exotic molecules, their spectra, and the unusual chemical processes that produce and destroy them under conditions often very different from those on our home planet. Chemical simulations involving thousands of reactions are now used to calculate concentrations and spectra of interstellar molecules as functions of time. Even biologists share an interest in the subject, because the interstellar clouds of gas and dust, portions of which collapse to form stars and planetary systems, contain organic molecules that may become part of the initial inventory of new planets and may indeed be the precursors of life. An irresistible subject to its practitioners, astrochemistry is proving to be exciting to a much wider audience. In this perspective article, the field is first introduced, and the emphasis is then placed on the three environments in which chemistry occurs in the interstellar medium: the gas phase, the surfaces of bare dust particles, and the ice mantles that cover bare grains in cold dense interstellar clouds. What we do know and what we do not know is distinguished. The status of chemical simulations for a variety of interstellar sources having to do with stellar and planetary evolution is surveyed. An optimistic view of the future of astrochemistry ends the article. Chemistry occurs on bare dust particle cores and on the ice mantles surrounding them via a variety of mechanisms.