The fall of the "Roman Empire" is most definitely the largest argument among historians.

For a contemporary account for cause of the fall, I will turn to Flavius Vegetius Renatus and his treatise on military affairs "De Re Militari" (written sometime in the 5th Century). Vegetius argues that the current (to him) influx of Germanic warriors into the Roman Army is resulting in the dilution of discipline and military readiness, and is also leading to increasing disloyalty of the Roman Generals (who are now also German). This theory is thrown out by modern historians. The only difference between the recruitment of "Barbarians" into the army from Vegetius' time and the time he calls back to is that now "barbarians" were being integrated into the frontline troops, where as before they were used solely as auxiliaries. However, there are no sources that point to proof that these "barbarians" were anymore disloyal or undisciplined than their more Roman counterparts. In fact two "Barbarian" generals, Flavius Aetius and Stilicho (more on both of these guys soon) were instrumental in the defense of Rome from the Huns. They were also dubbed by Edward Gibbon (more on him very soon) as "The Last of the Romans". So, they were hardly disloyal. Vegetius also points to the degrading equipment of the Imperial Army and blames that on the Germans, but the change in equipment had more to do with the fact that a lot of expensive steel was required for upkeep in the old equipment. The Empire simply couldn't afford an army the size they needed with the old equipment.

Edward Gibbon has, arguably, the most famous recounting of the fall of the Roman Empire from his 6 Volume "The History of the Decline and Fall of the Roman Empire" (1776-1788). His argument was that Christianity was the cause of the fall. According to him Roman society before the conversion to Christianity was focused on the "here-and-now", but after the conversion the populace was more complacent as they were all just waiting for heaven. This theory is almost completely thrown out by modern historians, who favor economic, military, institutional, and health problems as catalyst for the fall. It is also important to note that the Eastern Roman Empire was Christian and that Empire would last until 1453.

J.B. Bury's multi-volume "History of the Later Roman Empire" (1889-1923) deconstructs Gibbon. Bury argues that Gibbon's history was well researched and detailed, but that it focused too much on the "moral decay" aspect. Instead Bury states that a large combination of events in the 5th century resulted in the fall of the Western Roman Empire. He falls into the same trap as Vegetius in blaming the woes of the Roman Military on it's "Germanization" (which is a major simplification of his argument, read his books if you are interested in this argument more), but he also brings up three extra causes. He brings up economic decline (which we will get to more soon, I promise), but he also brings up the drama surrounding the two Generals I noted earlier, Flavius Aetius and Stilicho. First Stilicho was rumored to be involved in a plot to murder a man named Rufinus. Rufinus was the "man behind the throne" in the Eastern Roman Empire and the two men had a rivalry. Stilicho's rumored involvement in his death, and his failure to quell a revolt in Britain and Gaul lead to him being arrested and executed for treason. Flavius Aetius stayed around for 50 more years, but his sudden assassination in 453 meant that now the Roman Empire had no effective and loyal leadership in the west. According to Bury these events piled up leading to the Fall of the Western Roman Empire.

Peter Heather's "The Fall of the Roman Empire" (2005) also has a pretty similar theory to J.B. Bury; that multiple catastrophic events piled on towards the end of the Empire and brought it down. Heather argues that the Romans had to cope with the sudden rise of the Sassanid Persians. He argues that it took a half century for Rome to find a way to cope to the changing situation then the way that they ended up coping was flawed. According to him the Romans raised funds for their wars by taking all of the tax income from the west and moving it east. While this resulted in the stabilization of the borders in the east, it also led to some pretty nasty effects in the west. First, infrastructure projects became almost non-existent because there was no funding, and second, all of the well to do people moved from the outlying provinces to the Imperial Court (where the real money was.) So now we have a situation where the power is centralized in small areas, and the military is spread thin. What else can go wrong? Well the Germanic tribes had been growing stronger and stronger over the centuries of Roman contact. They were better educated, richer, and better organized than the had ever been. This became a catastrophic problem when the Huns began to move westward pushing the Germans into the Western Empire. Heather discounts Gibbon's belief that "moral decay" led to the fall, and he also discounts the belief that Roman infighting led to the fall. Heather argues that Roman infighting had gone on for the entirety of the Empire, and while it may have exacerbated the problems he listed, it can't be blamed for causing them.

Speaking of infighting though Adrian Goldsworthy's "The Fall of the West" (2009) argues that it was in fact Rome's constant civil wars and infighting that caused the fall. Goldsworthy goes on to state that the Army didn't become less effective, as it was still able to defeat it's enemies when it was prepared, but that instead the constant fighting between all of the major players led to a degradation of central authority, and political capacity (due to the constant assassination and overthrowing of effective leaders). This meant that the Romans didn't stand a chance when the social and economic problems became more and more apparent.

Finally there is the argument that Rome didn't actually fall in 476, but instead transformed. Chris Wickham's "The Inheritance of Rome: A History of Europe from 400 to 1000" (2010), Henri Pirenne's "Pirenne Thesis" (1920s), and Peter Brown's "The World of Late Antiquity" (1971) all argue that Rome didn't fall but instead the Germanic migrants and the Roman culture amalgamated and that the Carolingian Empire was the continuation of the Roman State. Wickham and Brown both point to how Medieval institutions and culture still had many roots in the old Roman Culture.

The modern consensus is that it is a combination of Goldsworthy, Heather, and Pirenne/Brown/Wickham, you can read more on that here. There a still criticisms of all of these beliefs, and there probably will be for the rest of time, but I certainly hope I answered your question well and that you are interested in the topic enough to check out some of these books. Take this list as my personal recommendation on good reading ;)