  A Killing $p$-form on a Riemannian manifold is a $p$-form whose covariant
derivative is totally anti-symmetric. In this paper we give the complete
(local) description of 4-dimensional Riemannian manifolds (M,g) carrying
non-parallel Killing 2-forms $\varphi$. If $M$ is connected and oriented, we
show that there exists a dense open subset of $M$ on which one of the three
exclusive situations holds: either $\phi$ is everywhere degenerate and $g$ is
conformal to a product metric, or $g$ is conformal to an ambik\"ahler metric
obtained via the Calabi construction from a polarized Riemannian surface, or
$g$ is conformal to an ambitoric structure of hyperbolic type, and depends
locally on two functions of one variable. We also give compact examples, by
constructing infinite-dimensional families of Riemannian metrics carrying
Killing 2-forms of each of the above types on $S^4$ and on Hirzebruch surfaces.
