We study anomalous heat conduction and anomalous diffusion in low dimensional systems ranging from nonlinear lattices, single walled carbon nanotubes, to billiard gas channels. We find that in all discussed systems, the anomalous heat conductivity can be connected with the anomalous diffusion, namely, if energy diffusion is $sigma^2(t)equiv <Delta x^2> =2Dt^{alpha} (0<alphale 2)$, then the thermal conductivity can be expressed in terms of the system size $L$ as $kappa = cL^{beta}$ with $beta=2-2/alpha$. This result predicts that a normal diffusion ($alpha =1$) implies a normal heat conduction obeying the Fourier law ($beta=0$), a superdiffusion ($alpha>1$) implies an anomalous heat conduction with a divergent thermal conductivity ($beta>0$), and more interestingly, a subdiffusion ($alpha <1$) implies an anomalous heat conduction with a convergent thermal conductivity ($beta<0$), consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our theoretical prediction.
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