  Motivated by the Bloch-Beilinson conjectures, Voisin has formulated a
conjecture about 0-cycles on self-products of surfaces of geometric genus one.
We verify Voisin's conjecture for the family of Todorov surfaces with $K^2=2$
and fundamental group $\mathbb{Z}/2\mathbb{Z}$. As a by-product, we prove that
certain Todorov surfaces have finite-dimensional motive.
