  We present a novel binding mechanism where a neutral Rydberg atom and an
atomic ion form a molecular bound state at large internuclear distance. The
binding mechanism is based on Stark shifts and level crossings which are
induced in the Rydberg atom due to the electric field of the ion. At particular
internuclear distances between Rydberg atom and ion, potential wells occur
which can hold atom-ion molecular bound states. Apart from the binding
mechanism we describe important properties of the long-range atom-ion Rydberg
molecule, such as its lifetime and decay paths, its vibrational and rotational
structure, and its large dipole moment. Furthermore, we discuss methods how to
produce and detect it. The unusual properties of the long-range atom-ion
Rydberg molecule give rise to interesting prospects for studies of wave packet
dynamics in engineered potential energy landscapes.
