How complicated is it to solve a degree 7 polynomial? How complicated is it to describe a branched cover, a field extension, a local system, a G-torsor? Much of topology was invented to address questions of this form, which are called resolvent problems. The numerical invariant which describes this notion of complexity is called the resolvent degree. I will discuss the history of resolvent problems, what's known about them (very little), and what's remains mysterious (a lot). I will also share a new sharp calculation in this area, the resolvent degree of the braid group. This is joint work with Claudio Gómez-Gonzáles and Jesse Wolfson.