I will survey on some long-standing open problems and some recent results about the regularity of minimizers of various relaxed energies. I will focus on the model case of harmonic maps from the 3-dimensional ball into the 2-dimensional sphere, as introduced in a celebrated work of Brezis-Coron-Lieb, and which was also widely studied in the context of liquid crystals. I will also mention some related recent results and open problems regarding fractional harmonic maps and Yang-Mills energies.