  This survey grew out of notes accompanying a cycle of lectures at the
workshop Modern Trends in Gromov-Witten Theory, in Hannover. The lectures are
devoted to interactions between Hurwitz theory and Gromov-Witten theory, with a
particular eye to the contributions made to the understanding of the Double
Ramification Cycle, a cycle in the moduli space of curves that compactifies the
double Hurwitz locus. We explore the algebro-combinatorial properties of single
and double Hurwitz numbers, and the connections with intersection theoretic
problems on appropriate moduli spaces. We survey several results by many groups
of people on the subject, but, perhaps more importantly, collect a number of
conjectures and problems which are still open.
