The nearest truly powerful radio galaxy, Cygnus A, is hosted by the central galaxy of a rich cluster, so that X-ray observations provide a physical probe of interactions with its environment. Deep Chandra observations of Cygnus A have revealed scattered clouds of relatively cool gas that are likely remnants of the former gas core disrupted by the jet. Detailed study of the cocoon shocks provides an estimate of the jet power. It also supports the case for approximately uniform pressure within the radio lobes, consistent with them being filled with relativistic plasma. X-ray imaging and spectroscopy reveal strong, diffuse inverse Compton emission from the lobes, seeded mainly by photons from the cosmic microwave background. This emission accounts for the small size of the X-ray “cavities” of Cygnus A compared to many less powerful radio galaxies hosted by a cluster central galaxy. A small X-ray hole found around hotspot E is evidence of the jet being deflected off the intracluster medium, back into the lobe and onward to the terminal hotspot.