  In this chapter, we present an overview of experiments with trapped Rydberg
ions and outline the advantages and challenges of developing applications of
this new platform for quantum computing, sensing and simulation. Trapped
Rydberg ions feature several important properties, unique in their combination:
they are tightly bound in a harmonic potential of a Paul trap, in which their
internal and external degrees of freedom can be controlled in a precise
fashion. High fidelity state preparation of both internal and motional states
of the ions has been demonstrated, and the internal states have been employed
to store and manipulate qubit information. Furthermore, strong dipolar
interactions can be realised between ions in Rydberg states and be explored for
investigating correlated many-body systems. By laser coupling to Rydberg
states, the polarisability of the ions can be both enhanced and tuned. This can
be used to control the interactions with the trapping fields in a Paul trap as
well as dipolar interactions between the ions. Thus, trapped Rydberg ions
present an attractive alternative for fast entangling operations as compared to
those mediated by normal modes of trapped ions, which are advantageous for a
future quantum computer or a quantum simulator.
