The first achievement of the Roman Empire was the numerous building/infrastructure many of these are still standing today one example of this is the Colosseum in Rome this was an amphitheatre used to host public entertainment e.g. gladiator fights.
The second achievement was the Julian calendar that was created by Julius Caesar. It is a simplified calendar as opposed to the ones that the Romans used. The calendar was for 365 days win an extra day every four years the calendar was used across Europe for over a millennia. Until Pope Gregory made a small adjust making the Gregorian calendar.
The third achievement was the Roman empire success in expanding. The Roman empire colonised North Africa, Britain, Spain etc. The infrastructure culture and roads influenced the people after the empire collapsed.
The fourth achievement was the Latin language. This was spoken throughout the empire and became the basis for modern European languages such as French Spanish Italian Portuguese English and romanian.
The fifth achievement was the spread of Christianity. The religion was adopted by Emperor Constantine who converted to Christianity in 312 AD. He then legalized Christianity and allowed Christians to worship openly. By the end of the Roman Empire most of the population were Christian.
The sixth achievement was the influence of the Roman army. The Roman army was very powerful and well organised. They had weapons like swords spears shields and helmets. Their tactics included using chariots and elephants. The Roman army conquered much of Europe and their tactics have been copied ever since.
The expansion of the Roman Empire was an achievement that had the greatest impact because even after the empire fell countries still had their infrastructure culture and religion.