Let \(X\) be the reduction of a Shimura variety (of PEL type) at a place of good reduction \(\mathfrak p\) of the reflex field. For a closed point \(x \in X\) denote by \(Z_{x,\ell}\) the Zariski closure of the \(\ell\)-power Hecke orbit of \(x\), where \(\ell\) is a prime number different from the residue characteristic \(p\) of \(\mathfrak p\). We explain the known results and conjectures about \(Z_{x,\ell}\), and the relation to the \(p\)-adic and \(\ell\)-adic monodromy for \(Z_{x,\ell}\).