I'm curious about how the Byzantine empire was viewed by contemporaries during the Middle Ages (11th century). By contemporaries, I mean the other powers of the era, such as Western European countries, Rome, etc.
Up to about the 11th century, the Byzantine Empire was seen as "king of the hill" that others wanted to knock down. That's because it straddled both Asia Minor and the Balkans in Europe, and incurred two sets of enemies.
Earlier, the Byzantine Empire managed to repel Arab attacks on the Dalmatian coast of the former Yugoslavia, in the 8th century (at a time when the Arabs were also crossing the Mediterranean to attack Spain). But a closer, fiercer enemy later arose in the Bulgarians, who managed to take most of the Balkans and threatened Constantinople itself.
To the east, the newly arrived Kievan Rus challenged the Byzantine Empire for the Black Sea. Also, the modern state of Georgia threatened to detach portions of the eastern part of the Empire.
Up to 1025, the Byzantine Empire was more successful than not, but after the 1025 death of its most capable ruler, Basil II, the empire reeled under progressively heavier blows from increasingly stronger enemies. After losing the battle of Manzikart in 1071 and being "caught" in the Crusades between roving western Europeans following the Schism of 1054 and the Saracens, in the latter half of the 11th century, the Byzantine Empire became a shadow of its former self