In this talk I will discuss one of the most classical objects of study in potential theory: equilibrium measures for the logarithmic energy. Given a compact set $E$, the (logarithmic) equilibrium measure on $E$ is the unique (if it exists) minimizer of the logarithmic energy among all probability measures supported on $E$. In the case of planar sets, the equilibrium measure coincides with the harmonic measure, and is rather well-understood. However, almost nothing is known about the equilibrium measures associated to subsets of higher dimensional Euclidean spaces. In a recent paper with Tuomas Orponen we show that these measures are absolutely continuous with respect to the arc-length measure on $C^{1,\alpha}$ curves in arbitrary dimension. I will describe some ideas of our proof, and mention many related open problems.