Modern day Italy was greatly influenced by ancient Rome. Ancient Roman ruins are most prevalent in Italy and the Italian language ( one of the 5 romance languages ) were based on the ancient Roman language of Latin. The current day capital of Italy is Rome, same as in ancient Rome. The Vatican, headquarters of the Roman Catholic Church is located as a separate country inside of Rome. This is so partly because ancient Rome made Christianity the official religion of the Roman Empire. For these reasons I believe that modern day Italy is influenced by ancient Rome more than other parts of the ancient Roman empire. This is of course an opinion. I do not think anyone can put forth a better one.
The art of the eastern and western parts of the Roman Empire were similar because in both parts of this empire it was Greco-Roman art. This is a term which indicates the fact that Roman art became heavily influenced by Greek art. Greek art had already influenced the eastern Mediterranean before it was taken over by the Romans because the Greeks had ruled this area for three hundred years prior to this. The Romans spread Greco-Roman art in the western part of the empire. It has to be noted that, however, many of the local peoples in the empire retained their traditional art.
Charlemagne increased the size of the Frankish kingdom to include most of the area of present France, Belgium, the Netherlands, Germany, and Switzerland, and parts of Austria, Italy, and Spain. It was the largest country in Western Europe since the Roman Empire, and much larger than any there today. On December 25, 800, Pope Leo III crowned him Emperor of the Romans, in an effort to reestablish the West Roman Empire. His country did not survive him for long, however, as it was divided among his grandsons. Nevertheless, the countries they began included France and the Holy Roman Empire. There are links to articles below.
I would say religion was the ancient Romans' most significant contribution to Europe. Christianity developed from a religion among a small group of Jews (who lived in Judea, which was part of the Roman Empire) into a mass religion in the Roman days. It spread around the Roman Empire. It became state religion. Catholic Christianity and Orthodox Christianity developed during the Later Roman Empire. They were originally called Latin or Western Christianity and Greek or Eastern Christianity respectively. The former was the main form of Christianity in the western part of the Roman Empire and the latter was the main form of Christianity in the eastern part of the Roman Empire.
The most famous early historian on the subject is Edward Gibbon who lived in the 18th century and who wrote the still-famous History of the Decline and Fall of the Roman Empire.
I think the question you are asking is why Charlemagne was crowned emperor of the Roman Empire. Though we do not have a really good record of why this happened, it is probable that the people involved wanted to reestablish the West Roman Empire and saw a chance to do so. First of all, the West Roman Empire did not suddenly collapse. People did not wake up some morning in 476 and see that everything had changed. For a long time after any date modern historians assign for the fall of Rome, there was the lingering idea in the West that the people who lived there were in the Roman Empire, even thought the government was not working. The West Roman Empire had decayed badly during the 3rd through 6th centuries. During that time, there were often a number of emperors at the same time, sometimes acknowledging the status of each other, and sometimes not. People got used to the idea that the Roman Empire was not defined by its emperors. When Germanic kingdoms were set up in the West Roman Empire, they were regarded as parts of the Empire, though not under really good control. Roman law was still in use in many places until well into the 8th century, and there was no general consensus that the Roman Empire was a thing of the past. In fact, the East Roman Empire was still one of the most important countries in Europe at the time of Charlemagne; today we call it the Byzantine Empire. The Franks had been building up their kingdom for a long time, and it had come to dominate the West even more than the Byzantine Empire dominated the East. Charlemagne had become King of the Franks, but was also, separately King of the Lombards. In fact, his power was arguably as great as that of any of the Roman Emperors of the West. There were a number of other things going on that argued in favor of reestablishing the West Roman Empire. Christians wanted a strong country to counter the Muslims in Spain. Pope Leo III had good reason to want Charlemagne to have as much power as possible, and in fact owed Charlemagne his life. There were people in the West to believed the empire should not be headed by a woman, and the East Roman Empire was, at that time, headed by Empress Irene; this might have influenced people at the time. So, Pope Leo III, possibly with the help of Charlemagne, and almost certainly with the help of some of Charlemagne's supporters, crowned him Emperor of the Roman Empire.
All of the lands which were under the Roman Empire were influenced by the Romans. Countries in the modern nation-state did not exist in antiquity. There were city-states, territories of ethnic groups which were named after the group and kingdoms.
Russia
Italy was the heart of the Roman Empire.
Russia
The Greek civilisation deeply influenced the Roman elite.
Iraq
The modern country that forms the southernmost border of the Roman Empire is Egypt.
It is a mixture of Greco-Roman and Egyptian-inspired architecture.
good question which i want to find out also
The most famous dictator of the Roman Empire
Russia
Mostly India and China. We don't know which one influenced the most, but seem to India since all temples that were built at that time dedicated to Hindu gods.