Visuomotor processing is selective - only a small subset of stimuli that impinge on the retinae reach perceptual awareness and/or elicit behavioral responses. Both binocular rivalry and attention involve visual selection, but affect perception quite differently. During rivalry, awareness alternates between different stimuli presented to the two eyes. In contrast, attending to one of the two stimuli impairs discrimination of the ignored stimulus, but without causing it to perceptually disappear. We review experiments demonstrating that, despite their phenomenological differences, attention and rivalry depend upon shared competitive selection mechanisms. These experiments, moreover, reveal stimulus selection that is surface-based and requires coordination between the different neuronal populations that respond as a surface changes its attributes (type of motion) over time. This surface-based selection, in turn biases interocular competition, favoring the eye whose image is consistent with the selected surface. The review ends with speculation about the role of the thalamus in mediating this dynamic coordination, as well as thoughts about what underlies the differences in the phenomenology of selective attention and rivalry.