Recent HAWC observations have found extended TeV emission coincident with the Geminga and Monogem pulsars. In this talk, I will show that these detections have significant implications for our understanding of pulsar emission. First, the spectrum and intensity of these "TeV Halos" indicates that a large fraction of the pulsar spindown energy is efficiently converted into electron-positron pairs. This provides observational evidence supporting pulsar interpretations of the rising positron fraction observed by PAMELA and AMS-02. Second, the isotropic nature of this emission provides a new avenue for detecting nearby pulsars with radio beams that are not oriented towards Earth. I will also discuss the implications of HAWC results for our understanding of the diffuse TeV gamma-ray and neutrino sky.