The Byzantine Empire is the term conventionally used since the 19th century to describe the Greek-speaking Roman Empire of the Middle Ages, centered on its capital of Constantinople. The Empire is also known as the Eastern Roman Empire, although this name is more commonly used when referring to the time before the fall of the Western Roman Empire. During much of its history it was known to many of its Western contemporaries as the Empire of the Greeks because of the dominance of Greek language, culture and population. To its inhabitants and its neighbors, the Empire was simply the Roman Empire.
At around 1000 AD, the Byzantine Empire had been weakened through Seljuk and Arab raids and invasions. Much land had been lost this way. A trader from China travelled down the Silk Road to Constantinople, bringing with him gunpowder. The Byzantines quickly adapted their new acquisition to fit their own needs.
In February 1001, the Seljuk Turks invaded part of northern Asia Minor. The Byzantines retaliated, utilizing their new bombs. The Seljuks retreated.
Two months later, a combined force of Arabs and Seljuks invaded the empire, but were again crushed by the advanced technology of the Byzantines.
The Empire grew once more, and the Byzantines quickly regained much of the land lost over the years. By 1032, the Byzantine Empire was greater than it had ever been.
In 1223, the Mongols under Subutai invaded Rus. The Byzantines, being a fellow Orthodox country, travelled north to help their allies. The Mongols were swiftly beaten back with the rudimentary muskets the Byzantines had developed.
The Mongols invaded Europe several more times, but the new Byzantine-Rus alliance defeated them every time.
The Byzantines, during the war with the Mongols, seized an large area along the Baltic coast, in OTL's Latvia, Lithuania and Estonia.
In 1492, Christopher Columbus sailed West to try to find a trade route to China, as all the conventional trade routes were blocked by the Byzantines. Columbus instead found the Americas. The Byzantines were quick to take advantage of the new world, and set up several colonies in Central and North America. In the Yucatan, they met with the Aztecs. But unlike OTL's Spanish, The Byzantines left the Aztecs alone, and set up successful trade systems with them.
By the 1600's, Constantinople was the most successful city in the world, and controlled much of the Americas.
In 1656, the Byzantine colony (OTL Mexico) rebelled, and gained independence. Many other colonies followed suite and by 1800, most of the Americas were free.
The Byzantine Empire flourished until 1656, when the revolution showed the world the Byzantines weren't unstoppable. Many parts of their vast empire rebelled, and the empire crumbled.
By 1798, the last vestiges of Byzantine rule disappeared forever. And a new Europe awoke.