The cosmic-ray flux of positrons is measured with high precision by the space-borne particle spectrometer AMS-02. A central open question is the origin of the positron flux above 10 GeV, which exceeds predictions for secondary production by cosmic-ray interactions with the interstellar medium. One leading explanation is that pulsars contribute significantly to this excess, a hypothesis strongly supported by observations of degree-scale TeV gamma-ray halos around several pulsars. These halos, produced by leptons from the pulsar doing inverse Compton scattering with the interstellar radiation fields, provide a powerful tool to study cosmic-ray transport around these sources. These TeV halos appear closely related to X-ray filaments observed around some pulsars, produced by synchrotron emission of the same leptons. In this talk, I will discuss how Galactic pulsars may explain the positron excess, what TeV halos reveal about particle propagation, and how particle-in-cell simulations help interpret these processes.