Another myth is that Rome fell due to "decadence," "debauchery," its "amoral" culture, or its persecution of Christians. It is simply baseless to compare Roman culture with ours, and decide its relative morality, then based on such a value judgement, say that Rome fell because it didn't meet a certain moral standard. Both of these explanations suggest that Rome was struck down by God — or some other supernatural force — and while you may believe it to be true, this can only be a matter of faith. Other, far more mundane and less supernatural explanations can be found for Rome's fall; we need not rely on value judgements such as this.
These Romans had a government based on both the Etruscan confederate tradition and the Greek "democracy" ideal. It was an oligarchy, whose chief body was the Senate. The Senate was made up of patricians (aristocrats) from the "original" tribes that had made up the population of Latium. The Senate was the Romans' version of what in a monarchy would have been an advisory council for a king. The Senate was led by a pair of consuls. The consuls presided over what eventually became an elaborate bureaucracy, the "executive branch" of the Roman Republic. The consuls wielded great power, as heads of the Senate and the bureaucracy — but each served a term of only one year, and thereafter could never hold that office. Also each consul could "veto" or countermand whatever the other did.
A secondary body, the Curiate Assembly, represented the "plebeians" or the common folk. While the Curiate Assembly had little legal authority, it represented the populace, and as such, greatly influenced the Senate and the consuls.
In addition to these bodies, Rome had an extensive bureaucracy. These offices controlled public works projects, adjudicated laws, and kept the peace. Over the years, as Rome's population grew, and began to exert ever more influence on the world about, the bureaucracy grew more complicated. Some offices could be held only by plebeians, some only by patricians; some held their offices for life, while others had terms of office which could be long or short.
The purpose of this form of government — which grew more complicated & cumbersome over the years — was to avoid what Rome had endured under the Etruscans: a monarchy. The Romans had hated the Etruscan kings who had ruled their city; once free of them, they set out to develop a system under which no one person would have too much power.
Finally, and perhaps most importantly, Rome had an army. Unlike many other states of the time, (with the exception of Sparta and a few other Greek city-states), this was a "standing army" of soldiers. Soldiering evolved into a profession, the older soldiers acting as commanders or officers. The need for their army was the potential for invasion by northern barbarians. A few centuries BCE, Celts had invaded Italy & sacked Rome. The possibility that the same thing might happen again, was very real. As Rome prospered, in the early days of the Republic, it became attractive to potential invaders. Only a standing army, ready to fend them off at any moment, could prevent another disaster.
With the defeat of Carthage, the military had proven its importance to Rome and the Romans began to dream of conquest. One by one, Greek cities either fell to them, or simply incorporated themselves into the Roman Republic. Elsewhere, Rome fought wars to capture and control distant lands and populations. These were taxed heavily, in order to support the military and to feed the needs of the folk of Rome. The need for a standing army, the renowned Roman legions, was justified, by these wars of conquest. They had a very intimidating reputation; in many instances, locals would not even fight them — the legions just marched in & their generals took over. Every successful conquest simply encouraged more of the same.
Conquering the known world was a costly escapade, and while it netted some benefits (which I'll get to in a moment), the cost was great, at home. The vast bureaucracy absorbed a good deal of the tax money collected from conquered lands, so the folk of Rome itself saw little of it. Furthermore, the city's population continued to grow. Shortages of food and water became commonplace, in spite of the complex system of aqueducts and roads that the bureaucracy had built. Riots resulted from popular discontentment.
Rome's vast bureaucracy, and the Senate, proved unable to do much about it. Their own system of government made it hard to get things done, since one man's order could be countermanded so easily by another. Senators and office holders jockeyed for influence — but as they played political games with each other, the true needs of Rome went unaddressed.
Inevitably, the only people with real power — military governors — took it upon themselves to attempt to solve the problem. Amid the last century B.C.E., three of them — Pompey, Marc Antony, and Julius Caesar — decided by common agreement to share control of the Republic. Together they directed the actions of the Senate and the consuls. Unfortunately, this power-sharing agreement, called the "triumvirate," fell apart, as the three grew suspicious of each other. Caesar was the first to reach Rome with forces in his command, and declared himself "imperator" or leader. He proved popular, and undermined the authority of his rivals, and for fear of his power, he was assassinated by a group of Senators.
Soon the Senate and the Curiate Assembly returned to their primeval roles as advisory councils, only this time to the Emperors, who were almost all generals and who used Rome's standing army to enforce their rule. They were dictators in the truest sense of the word, and most were brutal men who dealt with their rivals in the most severe ways.
Since their power came from the military, the early Emperors enlarged the Roman army, justifying this by launching offensives at the edges of the Empire. Many of these were poorly-planned escapades which ended in disaster. Even when gains were made, they were in places so distant that it was difficult to get reinforcements and occupation forces there — so many of the gains were soon lost.
All of this did little to help the situation at home in Rome — which had nominally been the reason for the switch from a Republic to a dictatorship. These distant campaigns weighed heavily on the people of Rome and soon it was clear that the Empire had reached its territorial limit. Emperor Trajan, therefore, made a momentous decision — that Rome had reached the extent that it could, and all expansion campaigns were halted. He and his successors went from conducting offensive warfare at the fringes of the Empire, to defending those borders.
Trajan's historic decision to halt offensive campaigns, however, stopped this steady stream of slave labor. Without enough labor to support it, Rome's economy slipped further. The infrastucture withered, without the labor to maintain it. Famine and drought reared its ugly head once more, and the Roman government, with a military dictator at its helm, could not deal with it.
Generals vied for power and there was a good deal of turnover in the office of Emperor. Occasionally, an able leader occupied that title, and was able to hold on to it, but even this was not enough to stop the turbulence.
At the edges of the Empire, outsiders applied pressure. They fought back and regained territory that Rome had claimed previously. With the collapse of Rome's economy, the Empire found it hard to defend its borders and stop these incursions. Inevitably, relentlessly, the barbarians that ringed the Empire chewed away at its edges.
Emperor Diocletian became convinced the Empire could not last in its current condition, so at the turn of the 4th century he divided it in two, an eastern and a western half. He kept the west, wherein Rome lay, and turned over the east to his protege Constantine. Constantine made the Greek trade town of Byzantium his capitol, and worked hard to shore up the Empire's middle eastern defenses.
Even so, the Roman legions still enjoyed a reputation as being extremely tough adversaries, & the barbarians, at first, were unwilling to penetrate too deeply into the Empire's territory, for in their own lands, the legions were considered unbeatable. This changed, however, with the Battle of Adrianople, the first occasion in which the legions were defeated, within the Empire proper. Their reputation was tarnished, ever after, & Rome never really recovered from this defeat.
The western half of the Empire, under more pressure from barbarians, and fewer resources with which to deal with them, collapsed under the constant invasions. The western Roman Empire was gone, by the start of the 6th century.
With large cities such as Antioch, Damascus, and Alexandria within it, the eastern Roman Empire had a greater capacity to defend itself, and did so. It lasted for nearly a thousand years, although for the last half of that period, it really consisted only of Greece and some Anatolian territory.
Briefly, under Emperor Justinian & his famed general Belisarius, the Byzantine Empire made an attempt to reconquer the west & restore the Empire. But, this campaign was fraught with trouble, and while Justinian & Belisarius had many successes (they took back most of Italy, much of Iberia, and many cities along the northern coast of Africa, and penetrated into modern Armenia), they were unable to accomplish this task. What was worse, most of these gains were lost within another century or so. The Byzantine Empire had reached its peak of power and extent in the 7th century; from then on, it was slowly worn away.
Might Rome have survived, had Trajan not chosen to halt the conquests? Possibly. However, the Roman government was inherently bloated and unstable. So it may have fallen in spite of that. Even so, we will never know. "What if's" have little value.