  In 2017, Ron Aharoni made the following conjecture about rainbow cycles in
edge-coloured graphs: If $G$ is an $n$-vertex graph whose edges are coloured
with $n$ colours and each colour class has size at least $r$, then $G$ contains
a rainbow cycle of length at most $\lceil \frac{n}{r} \rceil$. One motivation
for studying Aharoni's conjecture is that it is a strengthening of the
Caccetta-H\"aggkvist conjecture on digraphs from 1978.
  In this article, we present a survey of Aharoni's conjecture, including many
recent partial results and related conjectures. We also present two new
results. Our main new result is for the $r=3$ case of Aharoni's conjecture. We
prove that if $G$ is an $n$-vertex graph whose edges are coloured with $n$
colours and each colour class has size at least 3, then $G$ contains a rainbow
cycle of length at most $\frac{4n}{9}+7$. We also discuss how our approach
might generalise to larger values of $r$.
