We present the results of a deep imaging survey of the Virgo cluster of galaxies, concentrated around the cores of Virgo subclusters A and B. The goal of this survey was to detect and study very low surface brightness features present in Virgo, including discrete tidal features, the faint halos of luminous galaxies, and the diffuse intracluster light (ICL). Our observations span roughly 16 degrees2 in two filters, reaching a 3σ limiting depth of μB = 29.5 and μv = 28.5 mag arcsec-2. At these depths, our limiting systematic uncertainties are astrophysical: variations in faint background sources as well as scattered light from galactic dust. We show that this dust-scattered light is well traced by deep far-infrared imaging, making it possible to separate it from true diffuse light in Virgo. We use our imaging to trace and measure the color of the diffuse tidal streams and ICL in the Virgo core near M87, in fields adjacent to the core including the M86/M84 region, and to the south of the core around M49 and subcluster B, along with the more distant W cloud around NGC 4365. Overall, the bulk of the projected ICL is found in the Virgo core and within the W cloud; we find little evidence for an extensive ICL component in the field around M49. The bulk of the ICL we detect is fairly red in color (B - V = 0.7-0.9), indicative of old, evolved stellar populations. Based on the luminosity of the observed ICL features in the cluster, we estimate a total Virgo ICL fraction of 7%-15%. This value is somewhat smaller than that expected for massive, evolved clusters, suggesting that Virgo is still in the process of growing its extended ICL component. We also trace the shape of M87's extremely boxy outer halo out to ∼150 kpc, and show that the current tidal stripping rate from low luminosity galaxies is insufficient to have built M87's outer halo over a Hubble time. We identify a number of previously unknown low surface brightness structures around galaxies projected close to M86 and M84. The extensive diffuse light seen in the infalling W cloud around NGC 4365 is likely to be subsumed in the general Virgo ICL component once the group enters the cluster, illustrating the importance of group infall in generating ICL. Finally, we also identify another large and extremely low surface brightness ultradiffuse galaxy, likely in the process of being shredded by the cluster tidal field. With the survey complete, the full imaging data set is now available for public release.