  Afterglow radiation in gamma-ray bursts (GRB), extending from the radio band
to GeV energies, is produced as a result of the interaction between the
relativistic jet and the ambient medium. Although in general the origin of the
emission is robustly identified as synchrotron radiation from the
shock-accelerated electrons, many aspects remain poorly constrained, such as
the role of inverse Compton emission, the particle acceleration mechanism, the
properties of the environment and of the GRB jet itself. The extension of the
afterglow emission into the TeV band has been discussed and theorized for
years, but has eluded for a long time the observations. Recently the Cherenkov
telescopes MAGIC and H.E.S.S. have unequivocally proven that afterglow
radiation is produced also above $100$\,GeV, up to at least a few TeV. The
accessibility of the TeV spectral window will largely improve with the upcoming
facility CTA ({the} Cherenkov Telescope Array). In this review article, we
first revise the current model for afterglow emission in GRBs, its limitations
and open issues. Then we describe the recent detections of very high energy
emission from GRBs and the origin of this radiation. Implications on the
understanding of afterglow radiation and constraints on the physics of the
involved processes will be deeply investigated, showing how future
observations, especially {by} the CTA Observatory, are expected to give a key
contribution in improving our comprehension of such elusive sources.
