The 2D surface quasi-geostrophic equation (SQGE) has recently been a focus of much research. This equation appears naturally in geophysics.
It has a simple structure yet its solutions exhibit rich and complex behaviors. The SQGE is perhaps the simplest looking evolutionary equation of fluid dynamics for which the fundamental question of global existence of smooth solutions remains open. I will review some recent results on the regularity and roughening of solutions of the SQGE. In particular, I will talk about nonlocal maximum principle for SQGE, a new idea which found wider applicability, and about search for monotone quantities which can help understand energy transfer to higher modes.