The Roman Empire was a very powerful empire that ruled over much of Europe for many years. When it eventually collapsed, a lot of different people had to figure out what would happen next. This process is called succession. Basically, when the ruler of an empire dies or is overthrown, everyone has to decide who will be the new ruler, who will make sure everyone follows the same rules, and who will make sure everyone is safe. In the case of the Roman Empire, there were many different people who wanted to become the new ruler, and lots of fighting to try and decide who would take over. In the end, it was split into two parts. The western part of the Empire became what is known as the Holy Roman Empire, with its own set of rulers, and the eastern part became what is known as the Byzantine Empire, also with its own set of rulers. So, succession of the Roman Empire refers to the process of deciding who would take control of it after it fell apart.