Well there are two sections of the Roman Empire, the Western and the Eastern (which didn't end until 1527), but I'm going to assume you mean the Western. Rome was sacked multiple times throughout history, but the three most prominent times are by Visigoths in 410 AD, and the Vandals in 455 AD, and the Ostrogoths.
rome
If by Rome you mean the city of Rome, it was sacked by the Gallic tribe of the Senones in 390 BC. The Alemanni reached the outskirts between 258 and 260. Alaric I of the Visigoths besieged it twice and sacked it in his third attack (407-410). The vandals sacked it in 455. in 866 the arabs looted St Peter's, which was outside the walls. The Ostrogoths sacked it in 546 during the Gothic wars. In 1084 it was the Normans and in 1527 it was mutinous troops of emperor Charles V
Rome has been sacked by the Visigoths, Vandals, Ostrogoths, and many more.
Rome was sacked eight times: four times in antiquity and four times during the Middle Ages. It was sacked by the Gauls in 390 BC, by Alaric, king of the Visigoths, in 410 AD, by Ginseric, king of the vandals, in 455 AD and by Totila, the king of the Ostrogoths in 546 AD. In the Middle Ages the Vatican was sacked and the environs of Rome, including the Vatican, but the city of Rome itself was saved by her walls; The Normans sacked Rome in 1084. Finally Rome was sacked by the troops of Charles V, Holy Roman Emperor, during a mutiny in 1527. Alaric I besieged Rome and some slaves opened the gates. He plundered Rome for three days. Many great buildings were ransacked and many captives were taken. Tens of thousands of Romans let the ruined city and went to the countryside. However, the city then recovered. Ginseric promised Pope Leo I that he would not destroy the city or murder its citizens. The gates of the city were opened. It is accepted that Genseric looted great amounts of treasure from the city, and destroyed cultural objects (the term vandalism comes from this). There is some debate over the severity of the sack. Rome was plundered for 14 days. It is said that Ginseric did not burn buildings and that there was little murder and violence, and the Vandals did not burn the buildings of the city. However, a writer recorded that shiploads of prisoners to be sold as slaves arrived in Africa (where the vandals lived) from Rome. Therefore, the full extent of the sack is unclear. Some historians see this sack as 'the end of the Roman Empire', but this is an overstretched claim. Totila plundered the city, but did not destroy the fortification, even though he usually did so when he took a city. He withdrew and the walls and fortifications were retired quickly. He then marched on Rome again, but was defeated by a Roman general. The general was recalled from Italy and Totila advanced against Rome again and was let in through the treachery of its starving defenders and did not sack the city.
Aleric
The Gauls sacked and destroyed the city of Rome in 476.
Alaric and goths
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In 410 the Visigoths (Goths of the west) sacked the city of Rome. They did not destroy it.
Rome was sacked by the Visigoths in 410 and by the Vandals in 455.
vandals and visigothes
If you are referring to the city of Rome, as opposed to the Roman Empire, three Germanic peopled sacked Rome. The Visigoths in 410, the Vandals in 455 and the Ostrogoths in 546.
Although Rome was sacked 3 times in antiquity and 4 times in the middle ages, it was never captured.
No one sacked Rome during the Punic Wars.
The barbarians who conquered Rome were called the Goths.
Alaric I did not actually capture Rome. He sacked the city and left before the Roman army form garrisons around the empire would catch up with him. The sack was a shock because Rome had not been sacked for 700 years. It also showed that Rome was vulnerable.
The city of Rome was first conquered by foreign invaders in 410 AD, when the Visigoths, led by King Alaric, sacked the city. It was then again conquered in 455 AD by the Vandals, led by King Gaiseric.