Galois groups in enumerative geometry and applications

As Jordan observed in 1870, just as univariate polynomials have Galois groups, so do problems in enumerative geometry. Despite this pedigree, the study of Galois groups in enumerative geometry was dormant for a century, with a systematic study only occurring in the past 15 years. We discuss the current directions of this study, including open problems and conjectures.