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 ull. 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.
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