  We obtain a complete characterization of the bounded Hausdorff operators
acting on a Fock space $F^p_\alpha$ and taking its values into a larger one
$F^q_\alpha,\ 0 < p \leq q \leq \infty,$ as well as some necessary or
sufficient conditions for a Hausdorff operator to transform a Fock space into a
smaller one. Some results are written in the context of mixed norm Fock spaces.
Also the compactness of Hausdorff operators on a Fock space is characterized.
The compactness result for Hausdorff operators on the Fock space
$F^\infty_\alpha$ is extended to more general Banach spaces of entire functions
with weighted sup norms defined in terms of a radial weight and conditions for
the Hausdorff operators to become $p$-summing are also included.
