  Catanese surfaces are regular surfaces of general type with $p_g=0$. They
specialize to double covers of Barlow surfaces. We prove that the $CH_0$ group
of a Catanese surface is equal to $\mathbb{Z}$, which implies the same result
for the Barlow surfaces. We will also give a conditional application (more
precisely, assuming the variational Hodge conjecture) of the same method to the
Chow motive of low degree $K3$ surfaces.
