  The circumplanetary environments in our Solar System host a stunning array of
moon and ring systems. Study of these environs has yielded valuable insights
into planetary system formation and evolution, and there is every reason to
believe that we will have much to learn from the moons and rings that are
likely to exist in exoplanetary systems as well. This has motivated a small but
growing number of researchers to investigate questions related to the
formation, stability, long-term viability, composition, and observability of
these exomoons and exorings. Still, due to a range of significant observational
challenges, we remain at a relatively early stage of this work. As a result, we
continue to face a number of difficult, unanswered questions, but this also
means there are myriad opportunities for fundamental contributions to the
field. In this review we will examine a variety of important issues for the
astronomical community to consider, with an aim of providing a comprehensive
understanding of ongoing efforts to identify and characterize exomoons and
exorings, while also increasing interest and engagement. We begin with an
overview of what we expect from systems hosting moons and/or rings in terms of
their architectures, habitability, and observational signatures. We then
highlight the contributions from a variety of works that have been aimed at
detecting and characterizing them. We conclude by examining the outlook for
finding these objects and discussing a number of ongoing challenges that we
will want to overcome in the years ahead.
