  The smooth (resp. metric and complex) Nielsen Realization Problem for K3
surfaces $M$ asks: when can a finite group $G$ of mapping classes of $M$ be
realized by a finite group of diffeomorphisms (resp. isometries of a Ricci-flat
metric, or automorphisms of a complex structure)? We solve the metric and
complex versions of Nielsen Realization, and we solve the smooth version almost
completely for involutions. Unlike the case of $2$-manifolds, some $G$ are
realizable and some are not, and the answer depends on the category of
structure preserved. In particular, Dehn twists are not realizable by finite
order diffeomorphisms. We introduce a computable invariant $L_G$ that
determines in many cases whether $G$ is realizable or not, and apply this
invariant to construct an $S_4$ action by isometries of some Ricci-flat metric
on $M$ that preserves no complex structure.
  We also show that the subgroups of ${\rm Diff}(M)$ of a given prime order $p$
which fix pointwise some positive-definite $3$-plane in $H_2(M;\mathbb{R})$ and
preserve some complex structure on $M$ form a single conjugacy class in ${\rm
Diff}(M)$ (it is known that then $p\in \{2,3,5,7\}$).
