  We have conducted an imaging survey with the Hubble Space Telescope Wide
Field Camera~3 (WFC3) of 70 Galactic Cepheids, typically within 1~kpc, with the
aim of finding resolved physical companions. The WFC3 field typically covers
the 0.1 pc area where companions are expected. In this paper, we identify 39
Cepheids having candidate companions, based on their positions in
color--magnitude diagrams, and having separations $\geq$5$"$ from the Cepheids.
We use follow-up observations of 14 of these candidates with XMM-Newton, and of
one of them with ROSAT, to separate X-ray-active young stars (probable physical
companions) from field stars (chance alignments). Our preliminary estimate,
based on the optical and X-ray observations, is that only 3\% of the Cepheids
in the sample have wide companions. Our survey easily detects resolved
main-sequence companions as faint as spectral type K\null. Thus the fact that
the two most probable companions (those of FF~Aql and RV~Sco) are earlier than
type K is not simply a function of the detection limit. We find no physical
companions having separations larger than 4,000~AU in the X-ray survey. Two
Cepheids are exceptions in that they do have young companions at significantly
larger separations ($\delta$~Cep and S~Nor), but both belong to a cluster or a
loose association, so our working model is that they are not gravitationally
bound binary members, but rather cluster/association members. All of these
properties provide constraints on both star formation and subsequent dynamical
evolution. The low frequency of true physical companions at separations $>\!5"$
is confirmed by examination of the subset of the nearest Cepheids and also the
density of the fields.
