  This paper aims at the global regularity problem concerning the 2D
incompressible Boussinesq equations with general critical dissipation. The
critical dissipation refers to $\alpha +\beta=1$ when $\Lambda^\alpha \equiv
(-\Delta)^{\frac{\alpha}{2}}$ and $\Lambda^\beta$ represent the fractional
Laplacian dissipation in the velocity and the temperature equations,
respectively. We establish the global regularity for the general case with
$\alpha+\beta=1$ and $0.9132\approx \alpha_0<\alpha<1$. The cases when
$\alpha=1$ and when $\alpha=0$ were previously resolved by Hmidi, Keraani and
Rousset \cite{HKR1,HKR2}. The global existence and uniqueness is achieved here
by exploiting the global regularity of a generalized critical surface
quasi-gesotrophic equation as well as the regularity of a combined quantity of
the vorticity and the temperature.
