Large parts of the Western Empire, actually. The kingdom of Powys (later Wales) had a dynasty of rulers claiming descent from the Romans until the late 9th century, and until the English takeover in the 13th century royal authority was based on continuity. In France, particularly the south (Provence and Aquitaine), urban communities continued to be organised by the Roman 'senatorial class' (not the Roman senate, but the urban councils) who still self-identified as Roman, as opposed to Burgundian or Frankish, by the time of Gregory of Tours in the late 6th century. Their descendants would continue ruling the cities until the spread of Frankish and later French aristocratic rule by the 9th and 10th century. Before that time, the barbarians mostly simply formed a military upper layer in the hierarchy, with most local power networks continuing from Roman times. Barbarians (Franks, Goths, Burgundians, Lombards) might have been kings ruling over the kingdoms, but their populations (particularly the urban ones) were still predominantly Roman. This is best explained in Chris Wickham's work, both in Framing the Middle Ages and The Inheritance of Rome.