  Molecules containing short-lived, radioactive nuclei are uniquely positioned
to enable a wide range of scientific discoveries in the areas of fundamental
symmetries, astrophysics, nuclear structure, and chemistry. Recent advances in
the ability to create, cool, and control complex molecules down to the quantum
level, along with recent and upcoming advances in radioactive species
production at several facilities around the world, create a compelling
opportunity to coordinate and combine these efforts to bring precision
measurement and control to molecules containing extreme nuclei. In this
manuscript, we review the scientific case for studying radioactive molecules,
discuss recent atomic, molecular, nuclear, astrophysical, and chemical advances
which provide the foundation for their study, describe the facilities where
these species are and will be produced, and provide an outlook for the future
of this nascent field.
