While more can be said, check out this answer by /u/Libertat.

Part 1/2 (two parts needed due to character limit):

You’ve had some really great answers already on here (as well as on the previous answer by u/Libertat in particular). I’ve only been on this sub for a bit, and I’m super impressed by the quality of the responses.

I can try and add some more regarding what happened to the Roman aristocracy from the wider perspective of the socio-political configuration as a whole of the post-Roman kingdoms (specifically in the cases of Ostrogothic Italy and Vandal Africa), which should hopefully give a slightly broader view and add something to the answers already given.

Essentially, the post-Roman kingdoms of the west operated within what I would define as a Roman ideological structure (thus familiar to the aristocracy), but with a barbarian operational core (exemplified in military culture). Patrick Amory has written a very detailed ethnographic study in which he describes the synthesis of Roman and barbarian as rooted in the philosophy of civilitas - two ‘nations’ living in peace, yet performing different functions.

Whilst the Ostrogoths took operational command of the province, the traditional Roman political culture that the Italo-Roman aristocracy were used to very much prevailed in Italy - at least through the lens of our primary source for the Ostrogothic kingdom, the Roman statesman Cassiodorus. He wrote in ‘Romanised’ terms to the Eastern Emperor Anastasius essentially in order to prevent an Eastern strike on the (now fairly well-established) Ostrogothic state, stating that “Our royalty is an imitation of yours” and that “there are no causes for anger between us”. In addition, Theodoric the Ostrogoth’s palaces have also been observed to imitate the imperial style of Constantinople, Ostrogothic coinage usually carried the emperors bust, and Theodoric usually refrained from the imperial practice of minting gold coins (though this may have been due to merchants being reluctant to accept coins without the imperial bust more than anything else). In sum, the Ostrogoths didn’t want their polity to be seen as a rupture with Roman antiquity, but rather as if they were merely the new rulers of a province of the Western Roman Empire (in my view). The deferential posturing to the Constantinopolitan court as well as to the Nicene hierarchy (the Ostrogoths were Arian) prove this. This may have been out of practical necessity rather than an active choice, but the point that a ‘Romanised’ political culture persisted still stands.

A more practical example of sociocultural continuity in Ostrogothic Italy can be seen in the fact that while Theodoric issued edicts concerning Gothic law, he did not issue laws (leges) concerning his Roman subjects, instead promising to uphold civilitas - the established Roman civil law. This further meant that he was not infringing upon traditionally imperial prerogatives. Public building also continued, with Theodoric recorded in Cassiodorus’ Variae as stating “it is indeed our intention to build new things, but even more so to protect ancient things”. This fulfilled the traditional Roman conception of building projects embodying the idea of utilitas publica - usefulness to the state.

In the case of Vandal Africa, a continuity of political culture is evident in the location of the principal Vandal court within the old proconsular palace. The Vandal court s recorded to have hosted guests from across the Mediterranean, emphasising again continuity with the cosmopolitan political culture of the imperial past that the Roman aristocracy (united transregionally through paideia) would have been so familiar with. Roman civic institutions such as the flamines perpetuus (‘keeper of the imperial flame’) were kept alive in Vandal cities, but secularised as a symbol of civic pride as opposed to its pre-Christian symbolism within imperial religion. Traditions such as this “represented a strong centripetal force” (Andy Merrills and Richard Miles, The Vandals) due to the tangible link they forged with the Romano-African elite. In relation to legal culture: the Vandal King Geiseric’s will took the form of a Roman fideicommisum (one of the most popular instruments of Roman law), and Vandal court documents appear to refer to Vandal territories as ‘provinces’. The Albertini Tablets (a collection of cedar tablets concerning transactions on a Vandal estate owned by a flamines perpetuus) further demonstrate that Roman legal norms were established in everyday Vandal life. Pierre Riché has furthermore demonstrated that educational instruction continued in Vandal Africa; as the poet Dracontius recalls his education in the early Vandal years, Romans and Vandals were educated together in his class. This was accompanied by a high level of professional training, as evidenced by large numbers of medieval and legal writings. Research by Arrhenius into material culture has also shown that the cloisonné jewellery of the post-Roman elite had their origins in Roman regalia. All of these elements of continuity worked to the advantage of the Vandal state, primarily by placating the established Romano-African elite and conveying both an image and a practical reality of faithfulness to the ideals of Romanitas. Nevertheless, criticism made itself known, with Christian moralists claiming that the Vandal conquest marked the end of North African urban civilisation; Bishop Quodvultdeus of Carthage asked “Where is Africa, that for the whole world was like a garden of pleasures” (incidentally, evidence from the Theodosian Code as highlighted by Merrills and Miles notes that urban contraction had been a problem since the second half of the fourth century, with Constans II having to intervene). However, it should be emphasised that urban decline was a reality and, as Anna Leone as noted, the gradual disappearance of civic fora had broader political implications - being associated with the decline of the municipal authorities which once ruled them.

A further point in relation to ideological continuity under the Vandals: though King Geiseric’s sack of Rome in 455 seems on the surface to be an act of grave insubordination to Roman hegemony, we must remember that this took place after the usurpation of the western imperial throne by Petronius Maximus and the marriage of his son to the princess Eudocia, daughter of the usurped western Emperor Valentinian III. Thus, Geiseric framed the invasion as occurring in accordance with the princess’ request, whom he then married to his eldest son Huneric. This ideologically significant infusion of Theodosian blood into the Vandal line even came at the complete cost of Vandal-Visigothic relations (Geiseric annulled Hunderic’s marriage to Theodoric the Visigoth’s daughter and sent her mutilated body back to the Visigothic court). As Jonathan Conant writes (in Staying Roman: Conquest and Identity in Africa and the Mediterranean 439 - 700), this connection was recognised long after the end of Theodosian rule and gave the Vandal kingdom in Africa “a kind of legitimacy it would otherwise have lacked”. The prestige of the marriage was hailed by the sixth century Latin poet Luxorius: “Mighty Vandalric, heir of a twin crown, you have adorned your name through momentous deeds […] the great manliness of Valentinian […] is exhibited in the wile of [his] grandson”

There is obviously a ton of insanely good information being shared in this thread, but I thought I'd add one very important point that some replies appear to be missing. While the fate of wealthy Roman families in Rome remains an ongoing question, the broader truth seems to be that most aristocratic Romans left sometime in the 5th century. Studying the demography of the ancient city is really tricky business, but archaeological evidence suggests that the vast majority of wealthy, private residences were abandoned and buried sometime between the sacks of 410 and 455. This was as much due to shifting political and economic circumstances as it was the traumatic effects themselves of 410; actually, we don't see a ton of evidence of widespread destruction following 410 (and indeed see a fair amount of rebuilding), although some major exceptions exist. That being said, the political poles of the empire were majorly shifting, and the east was emerging as the more stable, powerful, and well-connected swathe.

What we can say for sure is that by the end of the 5th century, Rome's population had probably declined to minuscule levels compared to just a 100 years before. And while we see some limited evidence of ongoing occupation by lower class inhabitants, these are distributed in a sort of "leopard print" distribution across the city, with broad areas of the urban fabric abandoned. For another couple of hundred years after this, we don't have much evidence of aristocratic/wealthy housing in the city. A tantalizing glimpse might be offered by the nice pavements recovered on the Aventine from the 7th century, but it isn't until the 9th/10th with the houses in the imperial fora and Largo Argentina that we see a development of new "aristocratic" housing forms.

In sum, Rome was no longer the sensible or safe place to be for wealthy, powerful families, many of whom probably moved to Constantinople. For a general discussion of these issue (sorry I cannot provide a good English source):

Meneghini, Roberto, and Riccardo Santangeli Valenzani. 2004. Roma Nell’Altomedioevo: Topografia e Urbanistica Della Città Dal V al X Secolo. Archeologia Del Territorio. Roma: Libreria dello Stato, Istituto poligrafico e Zecca dello Stato.

What about the eastern roman empire aristrocrafic families? Do we know what happened to them?