A group $G$ is Jordan if there exists a constant $C$ such that
any finite subgroup of $G$ has an abelian subgroup of index at most $C$.
For example, $\mathrm{GL}(n,\mathbb{R})$ is Jordan for every $n$. Some 30 years ago E. Ghys
asked whether diffeomorphism groups of closed manifolds are Jordan.
A number of papers have been written on this question in the past few
years. It is known that there are lots of manifolds whose
diffeomorphism group is Jordan, and also lots of manifolds for which
it is not. However, Ghys's question is far from being completely
understood, and many basic and interesting problems related to it
remain open.

In this talk I will survey the recent developments and some of the
open questions about Jordan property for diffeomorphism groups.