Byzantine Architecture is one of three major forces in the architectural world during the Middle Ages of Europe. Chronologically, the Roman Empire fell in 476 AD. This date marks the beginning of the Middle Ages. However, before its fall, the empire was divided into the Eastern and Western Roman Empires. In 476, the "Western Roman Empire" fell, while the Eastern Roman Empire, whose capital became Constantinople, preserved Roman culture (and architecture) and became the Byzantine Empire.
Though this civilization wasn't the first to concentrate their architecture on religious themes, it was a strong feature among Byzantine architects (as with any aspect of medieval society). Most works were to glorify the Church, in this case, the Eastern Orthodox Church, or the Emperor. The best example is the Hagia Sophia, completed in 537. The Hagia Sophia (which means Holy Wisdom) was a Christian church at Constantinople (now Istanbul) at first. It was mosque under the control of the Ottoman Empire. Presently it is a magnificent museum. As a massive and ornamental church, the building represents the glory of the Byzantine Empire.
The religious buildings and their designs are the first achievements of the Byzantine Empire. Another important advancement was the development of bacilicas. Bacilicas were early Christian or medieval churches. This style was common in Roman Catholic and Eastern Orthodox churches.
Byzantine and Islamic architecture share a common trend: that is, the use of the dome. One example is the Dome of the Rock in Jerusalem, which was Islamic architecture, but illustrates the influence Byzantine bestowed as the dome style passed on to the Muslims. They often used "Persian' dome. We modernly refer to this as the Onion Dome. The most celebrated example is the Taj Mahal (A.D. 1630) at Agra, India. Byzantine's advancement in developing the dome created a new style in global architecture, for no other civilization had designed buildings, especially religious buildings, as did the Byzantines.
As you have seen, aspects of cultures may be passed from area to another through some method of cultural diffusion - either, war, trade, dominance, or some other means. The Byzantine Empire was a very influential Eastern power. In fact most of modern Russian culture is a reflection of Byzantine culture. Russians, along with many other Eastern nations exhibit architecture similar to that of the Byzantine Empire. One example is the Onion Domes that appear on the Kremlin in Moscow, Russia [below]. Another is the Assumption Cathedral in the Kremlin, Moscow (begun in 1475). These onion domes are prevalent in many kremlins throughout Russia as they were in the Byzantine Empire.