  We prove that for any closed smooth $4$-manifold $X$ there exists a constant
$C$ with the property that each finite subgroup $G<Diff(X)$ has a subgroup $N$
which is abelian or nilpotent of class $2$, and which satisfies $[G:N]\leq C$.
We give sufficient conditions on $X$ for $Diff(X)$ to be Jordan, meaning that
there exists a constant $C$ such that any finite subgroup $G<Diff(X)$ has an
abelian subgroup $A$ satisfying $[G:A]\leq C$. Some of these conditions are
homotopical, such as having nonzero Euler characteristic or nonzero signature,
others are geometric, such as the absence of embedded tori of arbitrarily large
self-intersection arising as fixed point components of periodic
diffeomorphisms. Relying on these results, we prove that: (1) the
symplectomorphism group of any closed symplectic $4$-manifold is Jordan, and
(2) the automorphism group of any almost complex closed $4$-manifold is Jordan.
