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 also produced 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, demonstrating how future observations, especially by the CTA Observatory, are expected to give a key contribution in improving our comprehension of such elusive sources.