The spiking response of a primary visual cortical cell to a stimulus placed within its receptive field can be up- and down-regulated by the simultaneous presentation of objects or scenes placed in the "silent" regions which surround the receptive field. We here review recent progresses that have been made both at the experimental and theoretical levels in the description of these so-called "Center/Surround" modulations and in the understanding of their neural basis. Without denying the role of a modulatory feedback from higher cortical areas, recent results support the view that some of these phenomena result from the dynamic interplay between feedforward projections and horizontal intracortical connectivity in V1. Uncovering the functional role of the contextual periphery of cortical receptive fields has become an area of active investigation. The detailed comparison of electrophysiological and psychophysical data reveals strong correlations between the integrative behavior of V1 cells and some aspects of "low-level" and "mid-level" conscious perception. These suggest that as early as the V1 stage, the visual system is able to make use of contextual cues to recover local visual scene properties or correct their interpretation. Promising ideas have emerged on the importance of such a strategy for the coding of visual scenes, and the processing of static and moving objects.