Wikipedia isn't the best source obviously but gives a good overview of both.
Demand from the Islamic world dominated the slave trade in medieval Europe. For most of that time, however, sale of Christian slaves to non-Christians was banned. As a result, most Christian slave merchants focused on moving slaves from non-Christian areas to Muslim Spain, North Africa, and the Middle East, and most non-Christian merchants, although not bound by the Church's rules, focused on Muslim markets as well. Arabic silver dirhams, presumably exchanged for slaves, are plentiful in eastern Europe and Southern Sweden, indicating trade routes from Slavic to Muslim territory.
https://en.wikipedia.org/wiki/Slavery_in_medieval_Europe#Slave_trade
Slavery in the Byzantine Empire was widespread and common throughout its history. The military campaigns and expansion of the empire in the 10th century resulted in a large numbers of slaves. The Synopsis of Histories mentions that after the Battle of Adrassos (960) many prisoners of war were sent to Constantinople. They were so numerous that they filled all the mansions and rural regions. After the 10th century the major source of slaves were often Slavs and Bulgars, which resulted from campaigns in the Balkans and lands north of the Black Sea. At the eastern shore of the Adriatic many Slav slaves were exported to other parts of Europe.
https://en.wikipedia.org/wiki/Slavery_in_the_Byzantine_Empire