Keywords: fractional Laplace, global regularity, the surface quasi-geostrophic equation Abstract:Fundamental mathematical issues concerning the surface quasi-geostrophic (SQG) equation have recently attracted the attention of many researchers and important progress has been made. This talk focuses on the existence, uniqueness and regularity of solutions to the SQG equation and covers both the inviscid and dissipative cases. We will summarize some existing work and report very recent numerical and theoretical results.