The idea that the western Roman empire fell due to increased luxury and decadence was very popular in the 18th-19th centuries, and continues to be cited in popular history documentaries today. It's also an idea that pops up among political commentators, who constantly make badly thought out comparisons between modern America and ancient Rome for whatever their agenda is. Theories of decadence could take various forms; Edward Gibbon famously posited that the adoption of Christianity by the Roman state gradually eroded the Roman military virtues, leading to the weakening of the Roman state and its eventual collapse (this is of course an oversimplification of Gibbon's argument). Others have taken the increasing reliance on "barbarian" armies to defend the empire as proof of the increasing decadence of the Romans, and usually cite colorful anecdotes about the luxurious lifestyles of the Roman emperors and the aristocrats of the empire, or note the increasing luxury of the 4th century AD, which was an age of affluence. Proponents of these decadence theories could also turn to moralizing ancient historians of the 5th and 6th centuries writing of the decline of Rome, and most notably the "pagan" historian Zosimus, who saw Constantine's conversion to Christianity as a turning point in the empire's fortunes.

However, there are many problems with these ideas and I don't know of any modern scholar who follows them (though they are still popular, as I said, in many documentaries and superficial treatments of the fall of Rome). For one thing, the ancient Romans had been complaining of increasing decadence and luxury essentially as far back as we can go in the reliable historical record. The second century B.C. politician Cato the Elder constantly complained of the increasing luxury of the Romans and his opponents (for example, Plutarch, Cato Maior 3) and claimed that Roman military virtue was declining as a result. The Augustan historian Livy also saw the early second century B.C. as a turning point for Rome's decadence (39.6), and Sallust, writing during the civil war period after the assassination of Caesar, believed that it was Sulla who in the early first century had corrupted the Roman character by bringing eastern luxury to Italy (The War Against Catiline 11).

The lesson here is that the Romans were constantly apprehensive about the effects of luxury on their society and were paranoid that the Roman "character" was declining due to decadence. In the 2nd century A.D., historians like Tacitus and Suetonius would write about the decadence of the Julio Claudian emperors, such as Caligula and Nero. A mysterious biographer, probably writing in the 4th century A.D., wrote biographies of "decadent" and lavish second and third century emperors, such as Commodus and Elagabalus. If decadence was a serious reason for the collapse of the western Roman empire in 476 A.D., then, why did it take over six centuries for this problem to actually have an effect on the Roman empire? Popular documentaries these days that claim that decadence was responsible for the fall of the Roman Empire will often point to figures like Nero or Elegabalus; but how could figures living two to four centuries before the fall of Rome have had a serious effect on the empire's collapse?

Similarly, if decadence was responsible for the west's collapse, why did the eastern Roman empire, based out of Constantinople, survive? It actually prospered over the next century and a half or so (after 476), until a series of destructive wars with Persia and the Arab conquests diminished the extent of the empire and forced it to undergo serious changes in the seventh century (see John Haldon's book on 7th century Byzantium for this). Even so the Byzantine empire survived as a major power for another 400 years, and would only be destroyed in 1453, when the Ottomans captured Constantinople.

There's therefore little reason to think decadence played a serious role in the fall of the western Roman empire. If you want some more recent, accessible works on this topic, check out Peter Heather's Fall of the Roman Empire and Peter Brown's The World of Late Antiquity (which doesn't deal with the fall of the West in great detail, but makes a great case for seeing the 3rd-6th centuries as a time of great cultural innovation). Peter Brown's Through the Eye of a Needle is a very detailed and possibly too academic book for an introduction, but it is a phenomenal analysis of wealth, Rome and the church in this time period. And to entertain yourself, enjoy this long list of reasons that have been given for the fall of the Roman empire: https://www.utexas.edu/courses/rome/210reasons.html