The Byzantine Empire, much like the Roman Empire, faced a formidable array of external enemies. However, it was largely internal decay which destroyed both empires. The political and economic stability of the empire by 1000 A.D. led to two lines of development which combined to trigger a pair of interlocking feedback cycles that, in turn, eventually wrecked the empire. First of all, there was the free peasantry upon which the government depended for taxes and recruits. When the empire had been under constant attack, land had been a poor investment. But once stability started to return in the eighth century, many nobles looked greedily upon the farmlands controlled by the free peasantry. There was a constant battle as the nobles tried to get these lands and enserf the peasants. The government, seeing the free peasantry as the backbone of its economy and defence, did what it could to defend them. Basil II in particular fought long and hard to defend the peasants, but even he was unable to break the power of the nobles.
Secondly, and unfortunately for the peasants, not all emperors were strong or even concerned enough to defend the peasants. This was especially true after Basil II's death in 1025 when the empire was at its height and a strong military seemed less necessary. Therefore, a series of weak rulers with little military experience succeeded Basil. During hard times, such as famine, nobles would take the chance to dispossess the peasants. This wouild lead to the decline of the free peasantry and army, which in turn forced the state to rely more and more on expensive foreign mercenaries. This further increased the tax burden on the peasants, which caused more of them to lose their lands, leading to more reliance on mercenaries and so on.
This vicious cycle weakened the economy and tax base to the point where the Byzantines could not even afford to maintain their navy. Therefore, they asked such rising Italian city-states as Venice and Genoa to fight their naval battles for them. The price they paid was to lower and eventually eliminate the 10% import toll the Venetians and Genoese would normally pay. This allowed them to undersell Byzantine goods, which lowered government revenues from trade as well as ruining the tightly run guilds of Byzantine artisans and craftsmen. The even lower revenues forced the Byzantines to rely even more on the Italians, who then got an even tighter stranglehold on the Byzantine economy, thus repeating the cycle.
This also fed back into the first feedback cycle as the loss of money from lower tolls forced the government to raise taxes further and create an even greater burden for the peasants. The combined effects of these cycles led to growing internal decay within the empire and growing tensions with the Italian city-states who were taking over more of the empire's trade.
Along with these processes, events elsewhere were closing in on the Byzantines in the tenth and eleventh centuries. By 1070, a new and more aggressive enemy, the Seljuk Turks, had replaced the Arabs as the main Muslim threat to the Byzantines. In 107l, at the battle of Manzikert, the Byzantines found out that, besides being expensive, mercenaries can also be unreliable. The result was a disastrous defeat when their Norman and Turkish mercenaries abandoned them without even fighting, leading to the loss of part of the Balkans and most of Asia Minor, the very heart of the empire. This, along with the declining economy described above, generated steady internal decay for the empire.
Desperate for help, the new emperor, Alexius I, made a plea to Western Europe for mercenaries. What he got instead was the First Crusade, a religious war with the goal of taking Palestine and Jerusalem from the Seljuk Turks. Alexius skilfully handled this wave of half civilized Westerners as they passed through his empire on the way to Palestine. He even managed to use them to recover part of Asia Minor. Alexius and his successors, John I and Manuel I did manage to stabilize the empire's frontiers and recover some ground. Unfortunately, in 1176, Manuel and his army were ambushed and severely defeated by the Turks at the battle of Myriocephalum. The lands regained over the last century were lost once again, showing how hollow the Byzantine recovery actually was.
Meanwhile, in addition to the Italian stranglehold on the Byzantine economy, growing cultural and religious differences led to rising tensions between the Byzantine East and Latin West. These tensions and the West's growing involvement in Byzantine affairs also helped lead to the First Crusade.
All the while, contact with the West kept growing, and with it friction between the two cultures. As the Italian city-states' stranglehold on the Byzantine Empire's trade grew, so did hostility against Italian merchants, who numbered some 60,000 in Constantinople alone. Cultural differences, such as how the two cultures carried on war and diplomacy, and a religious schism which split the Roman Catholic and Greek Orthodox Churches permanently in 1054, just added to the mutual animosity. In the late ll00's riots broke out in various Byzantine cities, causing the massacre of numerous Italian merchants.
A major backlash came from Western Europe in 1204 when Venice directed the soldiers of the Fourth Crusade to Constantinople, which they stormed and brutally sacked. A short lived Crusader state was set up but the Byzantines recaptured Constantinople in 126l. However, irreparable damage had been done. The Venetians still held strategic Aegean islands, and the Crusaders still controlled parts of Greece. Furthermore, much of the wealth and splendor of Constantinople had been hauled off to Venice and Western Europe.
The energy and resources the Byzantines used in recovering from this blow would have been better spent in meeting a potent new threat from the East: the Ottoman Turks. From 1300 onwards, the Ottomans steadily encroached on Byzantine lands in Asia Minor. In 1345 they crossed into Europe never to leave. The Byzantine state crumbled piece by piece into a pathetic remnant of itself. Finally in 1453, Constantinople, the last remnant of the old Roman Empire, fell to the Turks after a desperate and heroic siege. With that siege went the last remnants of the Roman Empire.