Byzantine Empire

The Eastern Roman Empire was the direct descendant of The Roman Empire, continuing its legacy into the Middle Age. In essence, it was the Greek-speaking Eastern part of Rome that was left standing after the barbarian invasions of the 5th century, prospering and surviving numerous ordeals until the Turks finally finished it off in 1453. The name "Byzantine" as we know today was coined, posthumously, by a German historian named Hieronymus Wolf in 1557, after the capital Byzantium, later Constantinople. The empire/kingdom called itself Roman to the very end, or in Medieval Greek: Βασιλεία Ῥωμαίων (Basileía Rhōmaíōn) "Roman Empire". The short form of the name was simply Ῥωμανία (Rhōmanía - not to be confused with that Romania); its inhabitants were "Rhōmaíōi", or Romans.

Other forms of its name include: Βασιλεία τῶν Ῥωμαίων (Basileía tôn Rhōmaíōn - "Empire of the Romans"), Ἀρχὴ τῶν Ῥωμαίων (Arche tôn Rhōmaíōn), Πολιτεία τῶν Ῥωμαίων (Politeίa tôn Rhōmaíōn - "Roman Republic"), and also Γραικία (Graikía) and Ῥωμαΐς (Rhōmaís). And of course, there was the obligatory Latin names: Imperium Romanum, Imperium Romanorum, Res Publica Romana.

Despite all that, the Empire was in no way Latin, but Hellenic. Greek, having been the language of Eastern Rome since antiquity, now took over Latin as the administrative language too. Greek Orthodox Christianity was the state religion, and high Romano-Hellenistic traditions of science and literature continued unbroken, though with a Christian overtone.

Also note that, as Byzantine and Rome were, in fact, one, any "founding" date of the medieval empire is completely artificial. There are two dates frequently used by modern historians: 324, the year Emperor Constantine I moved his capital to "New Rome" Constantinople, and 395, when the Roman Empire was formally divided into Western and Eastern halves.

For most of its lifetime, the Empire consisted of the Balkan peninsula and Asia Minor. Inheriting Roman military philosophy and discipline (and Conscription), Byzantium maintained, at least initially, a highly effective army based on heavy cavalry and professional infantry, and it was the undisputed naval power of Europe in The Low Middle Ages, with its infamous "Greek Fire", whose chemical components today still not known. Thanks to this, the Empire is often compared to a cork stopping Islam from pouring into Europe. As time went on, however, it lost more and more territory to foes, and its armed force became increasingly outdated and diluted.

Each Byzantine emperor naturally bestowed on himself the title of Roman Emperor, though this was greatly challenged by the Vatican, who insisted on calling him Imperator Romaniæ ("Emperor of Romania") but mainly Imperator Graecorum (Emperor of the Greeks) and the Byzantine empire as Imperium Graecorum, Graecia, Terra Graecorum or even Imperium Constantinopolitanum. Some notable lines of rulers are: the Justinian family, which saw the largest extent of Roman territory after the fall of the West, the Macedonian dynasty, boasting Basileios II "Bulgar Slayer", the Komnenoi, with Manuel I Kommenos and finally the Palaiologoi, which restored the kingdom from the mess caused by the Crusaders and defiantly carried on until the Fall of Constantinople. Men didn't have all the fun to themselves - there were powerful female figures as well, like Theodora and Irene of Athens. Political rivals had a curious tradition of mutilating opponent's face.

As the premier economy and culture of Europe, Byzantium was partly responsible for the Italian Renaissance, as scholars and craftsmen fled the shrinking kingdom in the 15th century. Romanesque and Ottoman architecture? Byzantium-inspired. Also, Byzantine silk was the first ever in Europe and very smexy - again, the Italians were their students and successors.

Speaking of succession, you might ask what has become of the Byzantine identity. Well, Greek people nowadays still consider Byzantium to be the medieval incarnation of Greece, some even go as far as demanding Istanbul, nay, Constantinople and Asia Minor back from Turkey. For a while, Russians saw themselves as the spiritual successors of Constantinople too, what with being Orthodox and all, would call Russia "Third Rome" and adopt many Byzantine symbols such as the double-headed eagle.

Tropes applicable to the Eastern Roman Empire:

 * Arch Enemy: As it turned out, the Seljuk and Osman/Ottoman Turks. Before that, the Mamluk Sultanate of Egypt. Before that, the Caliphate. Before that, Persia. It also had some really nasty wars with Bulgaria for a century or two.
 * Also the Papacy, Venice, and the Holy Roman Emperor also had their time as the Archenemy of Rome.
 * Art Evolution: Byzantine art endured through Late Antiquity and the Middle Ages. Though heading in an Oriental direction, it was very continuous and gave us the love for strong colours and intricate patterns that can still be seen in the aesthetics of Orthodox churches. And have you seen the Byzantine way of dressing reconstructed? Set quite the tone for Eastern European folk outfits, didn't it? All of that evolved more or less from the Roman tunic, with an exotic whiff from Persia.
 * Badass Decay: Towards the end. As more territory was lost, there simply wasn't enough economic base to gain back the wealth and power it once had.
 * Berserk Button: Khosrau II, emperor of Sassanid Persia, had the Empire on the ropes, winning battle after battle. Newly-crowned Emperor Heraclius was planning to ditch Constantinople and take up the less deadly job of ruling over the ruins of the probably doomed empire in North Africa. High on his own success, Khosrau sent Heraclius a letter, calling the Eastern Roman emperor his 'insensate slave' and then went on to taunt him by pointing out that Persia and its sun-worshiping religion was awesome and asked what good Jesus was if he couldn't even manage to save his own life? When the people of Constantinople heard this, they instantly transformed from whipped into extremely pissed off, Heraclius discovered a pair of testicles in his trousers, and beat the Persians so badly that, a few years later, Khosrau was assassinated and the burgeoning Muslim armies were able to wipe out the remnants of Sassanid Persia without even breaking a sweat.
 * Book Ends: Born by Constantine I, died with Constantine XI.
 * City of Spies: It wasn't called "Byzantine" for nothing.
 * Determinator
 * Deadly Decadent Court: As expected from a millennium-old political entity. We didn't get the word "byzantine" in English for nothing. (It's a negative stereotype held by Western Europe, granted, but still...)
 * The Empire / The Federation / The Kingdom: Depending on which side you cheer for, and the time period.
 * Eye Scream: A particular favorite method of dealing with rivals for the throne was having their eyes put out and shipping them to a monastery to spend the rest of their days.
 * Basileios the Bulgar-Slayer allegedly did this to more than 14000 prisoners of war after his victory in the battle of Kleidion. One in a hundred of them was allowed to keep one eye to lead them back to the Bulgarian Tsar Samuil, who is said to have suffered a heart attack from the sight of his men.
 * The Glory That Was Rome: Though having evolved into something else.
 * Hijacked by Jesus: The first in Europe, ironically.
 * To be more specific, what became known as Orthodox Christianity became the state religion of the Eastern Empire shortly after Constantine's death. By the Middle Ages, it had reached the point where it was intertwined with Byzantium itself.
 * Horny Vikings: Serving as the Praetorian Guard, called The Varangian Guard.
 * In Name Only: This is the viewpoint of the Catholic world. Byzantium's Western rivals were not inclined to consider it the legitimate heir of old Rome, but instead Imperium Graecorum, or Empire of the Greeks. Additionally, the Arabs, like the scholar Said Al-Andalusi, called the Byzantine Empire "the kingdom of the Greeks" and the Scandinavian Varangians described the country they served in as "Greece".
 * However, the Arab thing is complicate by the fact that one of the Arabic names for the Greeks, and particularly the Byzantine Greeks, was al-Rūm ("Rūm" is pronounced the way Americans say "room"), derived from "Roman". (Once the Arabs wised up to the fact that there was a linguistic difference between the Eastern and Western Empires, they began referring to the latter as al-Rūmān.)
 * Istanbul : Which was once Constantinople.
 * It Got Worse: The long decline of the Empire, but especially after 1204, when the Crusaders sacked Constantinople.
 * Last Stand: Made by the emperor himself during the final siege of Constantinople in 1453... nobody knows what happened to him., and a legend circulated that he had been turned into a statue and would return one day to reclaim the city.
 * Long Runner: As long as Ancient Rome itself.
 * Medieval Grome
 * Recycled in Space: The Roman Empire !
 * The Remnant: Even after Constantinople fell, a number of Byzantine remnants held out against the Ottomans for a few more decades, such as Mystras and Trebizond.
 * The Rival: To the Papacy and the Venetian Republic.
 * Vestigial Empire: One of the most glorious and enduring examples.

The Eastern Roman Empire in popular culture:
 * Byzantine is an essential faction in European medieval strategy games, like Age of Empires II, Crusader Kings and Medieval: Total War. In some Alternate History-capable ones, it's possible to have the Empire survive well into the 20th Century.
 * Similarly, history-based 4X and other Turn-Based Strategy games may have the Byzantines as a playable faction, as well. Civilization added the Byzantines in its third and fourth incarnations in their second expansion packs (Conquests and Beyond the Sword, respectively) under Theodora and Justinian, respectively.
 * Has a Fantasy Counterpart Culture in Harry Turtledove's Videssos and Guy Gavriel Kay's Sarantium.
 * I Am Skantarios
 * The Belisarius Series
 * In Axis Powers Hetalia, it's implied that eventually became Byzantium, which takes Turkey's role in ending the Empire a darker turn. This also could explain why Greece himself bears a grudge with Turkey.
 * In the Prince of Nothing series, the Nansur Empire is the Vestigial Empire of ancient Cenei, is ruled by a Deadly Decadent Court, lost control of its religion's holy city to heathen desert peoples and considers itself more cultured than the other Inrithi nations.
 * Byzantine stragglers attempt to retake Constantinople from the Ottomans in Assassin's Creed Revelations.