To any domain in the complex plane there can be associated a certain function of two complex variables, called the exponential transform. It first arose in operator theory in the 1970s, and has later been studied for its own sake by Mihai Putinar and myself. It turns out to have quite remarkable and useful properties. \vskip .1in \noindent More recently we have tried to generalize the exponential transform to higher dimensions (several real variables). The object we come up with, a kind of renormalized Riesz potential at critical exponent, turns out to have at least some interesting properties. I will give an overview of what we know so far.