Byzantine is a continuation of Roman Architecture yet it is more complex and uses bricks instead of stone, mosaics instead of carvings and domes instead of piers.
During 300 A.D. the Roman army started losing its power and with that, their engineering technologies. The fall of the Roman Empire after 300 and up to 700, was known as the Late Antique Age or the Early Middle Ages. 
The most important architectural style created by the Romans was the basilica. This style was used for the legal and commercial activities of the public.
Constantine introduced the Byzantium Empire in 312 and with it, came an architectural style inspired by basilica, which existed for the next 100 years.
The most distinct examples of Byzantine style architecture were built during the rein of Justinian I from 527 - 565 AD. His goal was to restore the Roman Empire within a Christian Context. It is believed that Justinian is responsible for bringing monumental architecture to Christianity.
His architects invented a system that transformed basic square churches into complex domes. Justinian developed a rule of law and reconstructed the Roman Empire, creating ramparts, civic buildings, residences, waterways, and churches.
One of Justinians churches was the octagonal-shaped Haiga Sophia. It was built to honor himself and his wife for protecting the territory from the barbarians. In order to afford such building, massive taxes were issued. Constantinople didn't react well to these ridiculous taxes, and resisted. Within a few days of disagreement, 30,000 men, women and children were left dead in the streets.
Use In Building Construction
Many innovations in dome construction occurred during the Byzantine era. Corbelling was the basic principal behind squinch and pendentive, both methods used to make the largest domes anyone had seen at the time. In order to create a corbel structure, each layer or structure overhung the previous, creating a backwards staircase appearance.A squinch is a construction filling in the upper corners on a square room to create an octagonal shape which is then corbled out into a dome. Pendentives developed when large domes became very popular in churches. They consists of a dome sitting on four corbelled arches.
The Byzantine Arch was developed and became very popular in Spain.
Mosaics were also developed during Byzantine ages. They were mostly used for religious images. 
Typical characteristics of Byzantine architecture include:
Brick and stone were the main building material for the walls and vertical supporting structure. Some of the building shad been very well preserved, yet others are nothing but ruins.
The roof was commonly composed of wooden trusses and tiled vaults. 
Gold mosaic tiles were glass with gold leafs pressed against the backside.
Examples of Byzantine Condominiums
One Hanson Place 1 Hanson Place, New York City
One Hanson Place was built in 1972 and was originally the tallest building in Brooklyn for over 80 years.
The buildings most distinctive feature is its four sided clock tower. The units range from studios to penthouse duplexes, all with large windows that allow occupants to enjoy the stunning views of Brooklyn.
Some of the features include 10 foot ceilings, hardwood floors limestone tiling, and modern kitchens. Amenities include a children's playroom, a fitness centre, bicycle storage and more.
Carlyle 35 East 76th Street, New York City, NY
Carlyle has a reputation for housing many famous residence. Some of which being the Prince of Wales, Marilyn Monroe, and Mick Jagger.
Many of the suites have individual layouts. There is a range of one to four bedrooms, with spacious living areas, full kitchens, and views over New York City. There are dining accommodations right in the building, convenient for all residence and guests.
Amenities include housekeeping, linen services, utilities (excluding phone), room service, concierge, doorman, valet, and a 24 hour fitness studio.