We present a deep imaging survey of the Virgo Cluster, designed to study the connection between cluster galaxies and Virgos diffuse intracluster light (ICL). Our observations span roughly 16 square degrees and reach a 3-sigma depth of mu(B)=29.5 and mu(V)=28.5 mag/arcsec^2. At these depths, the limiting systematic uncertainties are astrophysical: scattered starlight from foreground Galactic dust, and variations in faint background sources. The dust-scattered starlight is well-traced by deep far-infrared imaging, making it distinguishable from true Virgo diffuse light. Our imaging maps the Virgo core around M87 and the adjacent M86/M84 region, in subcluster B around M49, and in the more distant W cloud around NGC 4365. Most of the detected ICL is found in the Virgo core and within the W cloud, with little evidence for extensive ICL in subcluster B. The large amount of diffuse light seen in the infalling W cloud likely illustrates the importance of the group environment for generating ICL. The bulk of the detected ICL is fairly red (B-V=0.7-0.9), indicative of old stellar populations. We estimate a total Virgo ICL fraction of 7-15%, somewhat smaller than expected for massive, evolved clusters, suggesting that Virgo is still growing its ICL component. We trace M87s extremely boxy halo out to ~ 150 kpc, and show that the current stripping rate of low luminosity galaxies is insufficient to have built M87s outer halo over a Hubble time. Finally, we identify another large ultra-diffuse galaxy in Virgo, likely in the process of being shredded by the cluster tidal field.
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