The Chow ring generalizes in all dimension the group of divisors
modulo linear equivalence on a curve. Though its definition is quite
simple, it is a rather mysterious object, already for surfaces.
I will explain what is known and what is conjectured in this case. I will
conclude with an amusing result obtained with Claire Voisin, which sheds
some (tenuous) light on the Chow ring of K3 surfaces.