The Chow groups CH^i(X) of a complex projective manifold X have a rather complicated structure, in general, but one might hope that a for a prime p, the p-torsion and p-cotorsion subgroups are more easy to understand.
I will first survey the `classical' finiteness results for the p-torsion and p-cotorsion, and then discuss examples, based on joint work with A.Rosenschon, to show that outside this classical realm, the p-torsion and p-cotorsion are in general infinite. Our work builds on earlier works of Ceress, Nori, Bloch-Esnault and Schoen, which I will also discuss briefly.