Capernaum holds the strongest of the Galilean Gospel-site identifications, and a cautionary tale about the weakest.
The house of Peter. Beneath an octagonal Byzantine church, excavators found a first-century house that had been singled out very early: its main room was plastered while ordinary houses were not, pilgrims scratched prayers and the name of Jesus into its walls, and it was progressively enclosed, first as a house-church, then under the octagon that Byzantine builders reserved for major shrines. That is a traceable chain of veneration reaching back close to the events, which is as much as archaeology can offer for a site of this kind, and it is why the identification with the apostle Peter's house is taken seriously.
The synagogue that fooled everyone. A short walk away stands a handsome white limestone synagogue, long pointed out as the one where Jesus taught. The excavators established that it was in fact built centuries later, in the Roman or Byzantine period, but that it sits on the black basalt foundations of an earlier synagogue, which may well be the first-century building the Gospels describe. So the honest position inverts the tourist's: the impressive visible building is not the Gospel one, and the Gospel one, if it is here, is the humble foundation underneath.
Who holds it. The site is split between the Franciscans, who own and excavated the western half with the house and the modern memorial church, and the Greek Orthodox, whose red-domed church stands on the eastern half. The Franciscans admit visitors for a fee.