We discuss the influence on spin-fluctuation pairing theory of orbital selective strong correlation effects in Fe-based superconductors, particularly Fe chalcogenide systems. We propose that a key ingredient for an improved itinerant pairing theory is orbital selectivity, i.e., incorporating the reduced coherence of quasiparticles occupying specific orbital states. This modifies the usual spin-fluctuation theory via suppression of pair scattering processes involving those less coherent states and results in orbital selective Cooper pairing of electrons in the remaining states. We show that this paradigm yields remarkably good agreement with the experimentally observed anisotropic gap structures in both bulk and monolayer FeSe, as well as LiFeAs, indicating that orbital selective Cooper pairing plays a key role in the more strongly correlated iron-based superconductors. © 2017 American Physical Society.