In visual masking, visible targets are rendered invisible by modifying the context in which they are presented, but not by modifying the targets themselves. Here I summarize a decade of experimentation using visual masking illusions in which my colleagues and I have begun to establish the minimal set of conditions necessary to maintain the awareness of the visibility of simple unattended stimuli. We have established that spatiotemporal edges must be present for targets to be visible. These spatiotemporal edges must be encoded by transient bursts of spikes in the early visual system. If these bursts are inhibited, visibility fails. Target-correlated activity must rise within the visual hierarchy at least to the level of V3, and be processed within the occipital lobe, to achieve visibility. The specific circuits that maintain visibility are not yet known, but we have deduced that lateral inhibition plays a critical role in sculpting our perception of visibility, both by causing interactions between stimuli positioned across space, and also by shaping the responses to stimuli across time. Further, the studies have served to narrow the number of possible theories to explain visibility and visual masking. Finally, we have discovered that lateral inhibition builds iteratively in strength throughout the visual hierarchy, for both monoptic and dichoptic stimuli. Since binocular information is not integrated until inputs from the two eyes reach the primary visual cortex, it follows that the early visual areas contain differential levels of monoptic and dichoptic lateral inhibitions. We exploited this fact to discover that excitatory integration of binocular inputs occurs at an earlier level than interocular suppression. These findings are potentially fundamental to our understanding of all forms of binocular vision and to determining the role of binocular rivalry in visual awareness.