According to History.com, while the Eastern Roman Empire crumbled and fell, the western half led by Constantine I survived for another 1,000 years.
Byzantium, an ancient Greek colony for which the empire is named, is located on the strait that links the Black Sea to the Mediterranean Sea and served primarily as a place for transit and trade between Europe and Asia Minor. In 330 AD, Constantine declared a “new Rome” and made Byzantium into Constantinople, the new capital (which then became Istanbul, which is no one’s business but the Turks…).
The Byzantine Empire has a legacy because they left behind a rich tradition in art, literature, and philosophy which helped inspire the famous Italian Renaissance (which in turn inspired The Enlightenment, which in turn inspired The American Revolution, which in turn inspired The French Revolution). Remnants of Byzantine culture can be found Eastern Orthodox religion which is practiced in modern day countries such as Russia, Romania, Bulgaria, Serbia, and Greece.
History Channel also made a great documentary about the empire which you can view here: