In this talk, I will discuss a new global regularity result for axisymmetric, swirl-free solutions of the Euler equation in R⁴. These results involve significant differences with the classical global regularity results for the three dimensional case, because the transported quantity omega/r^d-2 may be unbounded even for smooth solutions in four and higher dimensions. Around the same time, Shao, Wei, and Zhang proved global regularity for axisymmetric, swirl-free solutions of the Euler equation in R^d for d=4,5,6, using somewhat different methods. I will discuss both of these results.