Human vision deals with two major limitations. First, vision is strongly foveated and deteriorates with eccentricity. Second, visual attention selectively prioritizes some stimuli over others. I review the functional and neurophysiological links between attention and eccentricity across a range of protocols including spatial cueing, crowding, dual tasking, and visual search, for both spatial and feature-based attention. The main conclusions are: 1) Attention and eccentricity are intrinsically intertwined. Functionally, attention partially compensates for peripheral vision's limits, effectively expanding the functional visual field. Neurophysiologically, attention appears tightly linked to receptive field properties across eccentricity and the visual hierarchy. 2) We must distinguish between attention as a mechanism and attention as an effect. Whereas its effects may vary, attention as a mechanism appears overall remarkably stable across eccentricity. 3) Eccentricity ranges have been severely limited. We know little about attention further into the periphery and findings beyond the effective oculomotor range have been ambiguous. 4) The spatial profile of the attentional distribution and how this is achieved remains to be determined, with evidence for gradients, Mexican hats, and rings. By default attention appears biased towards the center, but whether and in what way cognitive load aggravates this 'tunnel vision' remains unclear. 5) Research on feature-based attention as a function of eccentricity has been markedly underrepresented. The scarce findings suggest that it operates globally across the visual field, but to what extent it does so uniformly and how this changes the functional visual field remains unknown. Future empirical and modeling directions are suggested.