Arc Statistics
  The existence of an arc statistics problem was at the center of a strong
debate in the last fifteen years. With the aim to clarify if the optical depth
for giant gravitational arcs by galaxy clusters in the so called concordance
model is compatible with observations, several studies were carried out which
helped to significantly improve our knowledge of strong lensing clusters,
unveiling their extremely complex internal structure. In particular, the
abundance and the frequency of strong lensing events like gravitational arcs
turned out to be a potentially very powerful tool to trace the structure
formation. However, given the limited size of observational and theoretical
data-sets, the power of arc statistics as a cosmological tool has been only
minimally exploited so far. On the other hand, the last years were
characterized by significant advancements in the field, and several cluster
surveys that are ongoing or planned for the near future seem to have the
potential to make arc statistics a competitive cosmological probe.
Additionally, recent observations of anomalously large Einstein radii and
concentrations in galaxy clusters have reinvigorated the debate on the arc
statistics problem. In this paper, we review the work done so far on arc
statistics, focussing on what is the lesson we learned and what is likely to
improve in the next years.
