The Eastern Roman Empire or Byzantium was originally an integral part of, and eventually the continuation of the Roman Empire in its eastern provinces, after the fragmentation and fall of the Western Roman Empire in the 5th century A.D. 
Between 324-337 A.D., Emperor Constantine I the Great reorganised the empire establishing the new capital of the Roman Empire in Byzantium, and giving it his name - Constantinople in 330 A.D. (now known as Istanbul, in Turkey).

Constantine legalised Christianity, and Emperor Theodisius I endorsed it as the Empire's official state religion during his reign (379-395 A.D.). Then under the reign of Heraclius (610–641 A.D.), the empire's military and administration were restructured establishing Greek for official use, rather than Latin. Over the centuries, although the Roman state and its traditions were maintained, the empire eventually moved away from its Latin roots and oriented towards Greek culture instead, developing its own version of Christianity known as Eastern Orthodox. This shift is clearly reflected in  Byzantine coinage, as we shall demonstrate further on.

The Empire continued to exist for an additional thousand years, with its citizens referring to it as the Roman Empire, and to themselves as Romans. At its peak, it would become one of the most dominant and powerful forces of the era, boasting a rich culture, solid economy, and strong military tradition. In the 6th Century, Emperor Justinian I would reconquer and restore vast amounts of territories including Italy, the Balkans and most of the North African Coast. 

Most of these gains would be reversed by the end of the 9th Century. 
The period between 867-1180 A.D. would see a resurgence however, including the reconquest of various territories, but the inception of the Crusades and growing threat from other cultures, especially that of Islam, would eventually find Constantinople being sacked in 1204 A.D.
The next 150 years would see the Empire disintegrate, as civil wars and external threats took their toll.
The wind sown with the crusades, had now become a whirlwind, as a new dominant force surged over the horizon of world history, and eventually many a shore. Whatever was left of the Byzantine Empire was crushed by the Ottoman Turks with the inevitable fall of Constantinople in 1453.

As for the Maltese Islands, little is known about the period following the collapse of the Western Roman Empire and the transition to Byzantine rule. 
The first documented instance of Byzantine presence in Malta was in 533 A.D., in the form of general Belisarius, who landed here while on his way to North Africa. The islands were integrated 2 years later into the Byzantine Province of Sicily, with the main settlements remaining the city of Melita, and the Citadel site in Malta and Gozo respectively. 

The relatively high quantity of Byzantine ceramics found on the islands suggest that Malta might have had an important strategic role within the empire from the 6th to 8th centuries. 
The shift of the Byzantine Empire's culture and traditions from Latin to Hellenistic was reflected here too, with evidence suggesting the presence of a Greek Orthodox community, and Greek being adopted as the official / administrative language.  

Not much is known about the language spoken by the locals, but it was probably a dialect of Punico-Greco-Latin origins, with a renewed influx of Greek over time. This pattern is attested in social, artistic and other aspects of local culture including architecture.

A series of military campaigns would see the islands contested by various powers, even after the the Islamic take-over of the islands in the late 860s A.D.
This brought an end to Byzantine rule and by the beginning of the 10th century, Malta was under Arab control.

Despite some 335 years of Byzantine control, not much survives from this period in terms of buildings and structures but other archaeological evidence suggests a flourishing trade present between the 6th-8th centuries AD. The Byzantines were seasoned traders and masters of navigation.
Their merchant ships would have been a regular sight around our coast, and all across the Mediterranean.
To read more about the various aspects of Byzantine Culture in Malta, kindly access the links below