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The byzantine empire was at it's height during 565 under the leadership of Justinian. It included most of the lands around the mediterranean sea. Slowly after the Byzantine empire started to fall apart.
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Lombards invade Italy, eventually taking Northern Italy from the Byzantines. They never gained back that territory. This was one of the first steps to the downfall of the Byzantine empire.
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Muslims attack Constantinople. This was another stepping stone to the fall of the empire. they soon recovered from this.
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In 726 Emperor Leo III banned the use of Icons. He Banned these because he felt like people were focusing on the icons more than God himself. I believe this was for the better.
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In 1055 they suffered the loss of southern Italy to the Normans. They never gained this territory back. This was another stepping stone to their defeat
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The byzantine empire suffered a defeat at Manzikert to the Seljuk Turks. This caused a permanent loss of most of Asia Minor.
This was a mojor setback -
In 1096 crusaders arrive at Constantinople. The Crusaders are successful, but eventually withdraw from cooperation with the Byzantines. So pretty much they got the territory back. Still this was another steppingstone to the downfall.
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Fourth Crusade captures Constantinople. The Latin Empire of Constantinople is formed as well as many Byzantine successor states. The capture of Constantinople in 1204 was a blow from which the Byzantines would never fully recovered.
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The successor state of Nicaea recaptures Constantinople and restores the Byzantine Empire.
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In 1453 the byzantine empire finally fell to the ground at the hands of The ottomans. The ottomans renamed this territory instabul. This was the ultimate defeat.