Discerning a target amongst visual 'clutter' is a complicated task that has been elegantly solved by flying insects, as evidenced by their mid-air interactions with conspecifics and prey. The neurophysiology of small-target motion detectors (STMDs) underlying these complex behaviors has recently been described and suggests that insects use mechanisms similar to those of hypercomplex cells of the mammalian visual cortex to achieve target-specific tuning. Cortical hypercomplex cells are end-stopped, which means that they respond optimally to small moving targets, with responses to extended bars attenuated. We review not only the underlying mechanisms involved in this tuning but also how recently proposed models provide a possible explanation for another remarkable property of these neurons - their ability to respond robustly to the motion of targets even against moving backgrounds.