The Byzantine Empire, known for its rich history and culture, never fully conquered Rome in the way one might think. After the fall of the Western Roman Empire in 476 AD, the eastern half, referred to as the Byzantine Empire, continued to thrive, with its capital in Constantinople.
However, there were moments when Byzantine emperors attempted to reassert control over Italy. Notably, during the reign of Emperor Justinian I in the 6th century, the Byzantine Empire launched military campaigns known as the Gothic Wars, which aimed to reclaim parts of the former Western Roman Empire, including Rome. In 536 AD, Byzantine general Belisarius captured Rome, but the city changed hands several times during the conflict with the Ostrogoths.
Ultimately, while the Byzantines did manage to occupy Rome temporarily, they never established permanent control over the city or the Italian peninsula. By the time of the prolonged conflicts and raids in the following centuries, the political landscape had changed significantly. The rise of the Papal States and various other powers contributed to the fragmentation of Italy, further distancing the region from Byzantine control.
In conclusion, while the Byzantine Empire conquered Rome at certain points, it did not maintain long-term control, and thus it can be said that they never truly conquered it in a lasting sense.