How nice are critical knots of knot energies? We already know that critical knots of the Möbius energy are smooth. This leads to the question whether critical knots of the Möbius energy are not only smooth, but also analytic. In this lecture, we give a short overview on the regularity results for the Möbius energy and present techniques with which the open question on the analyticity was solved. To the best of our knowledge, this is the first analyticity result in the context of non-local differential equations.