Any classical solution of the 2D incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component \(u_j\) of the velocity field \(u\) is determined by the scalar \(\theta\) through \(u_j =\mathcal{R} \Lambda^{-1} P(\Lambda) \theta\) where \(\mathcal{R}\) is a Riesz transform and \(\Lambda=(-\Delta)^{1/2}\). The 2D Euler vorticity equation corresponds to the special case \(P(\Lambda)=I\) while the SQG equation to the case \(P(\Lambda) =\Lambda\). We develop tools to bound \(\|\nabla u||_{L^\infty}\) for a general class of operators \(P\) and establish the global regularity for the Loglog-Euler equation for which \(P(\Lambda)= (\log(I+\log(I-\Delta)))^\gamma\) with \(0\le \gamma\le 1\). In addition, a regularity criterion for the model corresponding to \(P(\Lambda)=\Lambda^\beta\) with \(0\le \beta\le 1\) is also obtained.