In this talk, I will begin by giving some background on the joint moments of the first-order derivative of CUE characteristic polynomials, as well as the polynomials themselves, evaluated inside or on the boundary of the unit disk. I will then introduce some of my recent work on this topic and discuss its connections to Painlevé equations. Finally, I will list a few interesting and largely unexplored problems in this area.  This talk draws on collaborative work with Thomas Bothner, on some work with Nicholas Simm, and on additional collaborations with Theodoros Assiotis, Mustafa Alper Gunes, and Jon Keating.