The escape of cosmic rays from their sources and their subsequent propagation in the surrounding
medium remain central open issues in high-energy astrophysics, owing to their inherently multi-
scale and nonlinear nature. The recent discovery of TeV gamma-ray halos and X-ray filaments around
middle-aged pulsars provides direct evidence of particle escape and diffusion on scales of tens of
parsecs, at multi-TeV energies where direct CR measurements currently offer no constraints on
transport. So far, only three TeV halos have been firmly detected, but ongoing surveys are rapidly
expanding the sample of candidates. The inferred strong suppression of the diffusion coefficient in
these sources raises fundamental questions about its origin and implications for Galactic cosmic
ray transport, the positron fraction, and diffuse gamma-ray emission. In this contribution, we review the
current theoretical scenarios proposed to explain these observations—ranging from pre-existing
turbulence to anisotropic or self-generated diffusion. In addition, we discuss how X-ray filaments
may trace current-driven instabilities triggered by partially charge-separated escape of particles.
Future observations with LHAASO, the ASTRI Mini-Array, and CTAO, combined with kinetic
and MHD simulations, will be crucial to establish a consistent picture linking particle acceleration,
escape, and Galactic propagation.