The P-prime operator is a CR invariant operator on CR pluriharmonic functions and is closely related to a sharp Moser--Trudinger-type inequality in CR manifolds.  I will describe some analytic and geometric properties of this operator, and in particular use it to solve a nonlinear PDE of critical order which is the CR analogue of the Q-curvature prescription problem.  This talk is based on joint works with Paul Yang and Chin-Yu Hsiao.