This is a great question and has stimulated some interesting follow-ups in the other comments. I'll do my best to cover at least some of them.

Slavery at the height of the Roman Empire was truly historically unique. The Empire essentially functioned as a vast military-industrial complex, and the Roman elite used this system to amass a spectacular amount of social power and material wealth. Traditional slavery often functioned as domestic servitude, something along the lines of a live-in maid or field hand (albeit with limited legal rights and increased vulnerability to violence).

With the rise of the imperial elite, however, new forms of slavery became possible. A small tier of only about 600 households—the Senatorial elite—amassed resources for huge plantations and industrial centers (mines, pottery workshops, etc.), aggregating slave populations numbered in the hundreds and perhaps thousands. One study suggests that the top 1% of society owned the bottom 5%, or some 2.5 million souls. Perhaps a total of 5 million Roman subjects were enslaved, or 10% of the Empire.

Beginning in the 5th century, the Roman Empire began to politically fragment. The group most immediately affected was the elite stratum of society, those 600 households that held hundreds or thousands of slaves each. Particularly in the west, their ability to control these estates rapidly crumbled. North of the Loire River in France, villa estates and presumably the elites who managed them completely disappear.

In the west, slavery probably devolved to more traditional modes of domestic service. St Patrick is an excellent example from right around 400: a Roman elite captured by raiders and sold across the Irish Sea into slavery, where he worked as a shepherd for three years. This wasn't the chain-gang labor, nor does Patrick's owner seem to have managed a large estate. Surprisingly, Patrick seems to have been rather untroubled by the experience. Although he later spoke out against slave raiders who had attacked a baptism party, he never espoused any sort of abolitionist sentiment. Slavery—and slaving—continued to be an accepted part of life even after Rome fell.

In the east, of course, the Roman Empire didn't fall, although it severely contracted in the 600s. In Egypt and Syria, large slave estates probably survived transition from Roman to Islamic rule. In the decades leading up to the Islamic conquests, massive irrigation projects were still underway. Some of these represent private estates where large groups of slaves probably labored. Many elites were allowed to retain their properties, providing a tax base for the new rulers. The civil wars of the 600s and 700s would have disrupted estate management, but later evidence shows that large-scale slavery continued unabated. The massive palatial complexes of Samarra are one testament to their labors.

Fast forward to the 18th century. Slavery continued in some form in parts of Europe (particularly associated with Spain, Venice, and Constantinople) well into the colonial period. Spain brought her colonies legal codes for slavery that were indeed inspired by and perhaps even continuous with late Roman and Visigothic (the group that supplanted Roman rule in Spain) precedent. For Britain, France, and the Dutch, slavery had transformed into other forms of labor relationships, such as serfdom or indentured service. They certainly looked to Spain (and hence some Roman legal heritage) as one inspiration, but they also adapted freely to meet their own social, economic, legal, demographic, and environmental parameters.

So Roman law was valued to some extent in the development of early modern slavery, but the links were fairly tenuous. I'd argue that the Roman legacy was much more significant along two other dimensions: religion and culture.

First, the New Testament is peppered with anecdotes that preserve idea about Roman slavery: how slaves should be treated, how slaves should treat their masters, and how slaves should go about their work. Modern English translations often use the word "servant" but New Testament Greek uses an unambiguous word that means slave. This contributed to a broad cultural acceptance of slavery that went largely unchallenged until the 1800s. It was only during this period that Christianity first became firmly associated with abolitionism.

Second, Roman authors also wrote down ideas about slavery, which in some ways echo the New Testament. A good master was supposed to take paternal care of his slaves, treating them like children and disciplining them supposedly for their own benefit. Slave owners in the 18th and 19th centuries read the Classics as part of their education, and some sought out Roman manuals on running agricultural estates. Many of these modern slave owners adopted the same condescending paternalism as the elite Roman authors. In the US in particular, this condescension came to accompany slavery and has had lasting effects as one aspect of racism in America.

Selected sources:

• Jennifer A. Glancy, Slavery in Early Christianity (Oxford: Oxford University Press, 2002).
• Kyle Harper, Slavery in the Late Roman World, AD 275–425 (Cambridge: Cambridge University Press, 2011).
• Robin Fleming, Britain After Rome: The Fall and Rise, 400-1070 (London: Penguin, 2010).
• Micheal Decker, Tilling the Hateful Earth: Agricultural Production and Trade in the Late Antique East (Oxford: Oxford University Press, 2009).
• Enrico Dal Lago and Constantina Katsari, eds., Slave Systems: Ancient and Modern (Cambridge: Cambridge University Press, 2008).

I have a question similar to this. Around and during the fall of Roman civilization, were slaves treated differently then previously? Did they get full freedom once the empire fell? Or were they still oppressed even after?

A question based on your question, did slavery go away between the 5th and 18th centuries or something? I mean, I know there was the Encomienda system in the Spanish Colonies, but I also know that with feudal serfdom and indentured servitude, there would be less of a demand for 'slaves'. Was slavery a new thing, or just an expansion of an already widespread system?

If it already existed, was it questioned during any major social reform, or just kinda put aside?

Follow-up question. If someone wants to explain the logistical / institutional issues, I would not mind knowing more. What happened to the slaves in lands invaded by the "barbarians" for example?

The slavery persisted in Nordic countries probably until 13th century. Träldom was officially abolished 1335 by king Magnus Ericsson of Sweden, Norway and Skåne. I believe slavery had little practical meaning at that time, the catholic church had preached for centuries that only pagans could be enslaved and the thralls had been baptised and manumitted. The pagan countries on Baltic coast were forcibly converted in previous century.

I'm no expert in English history, but I believe the slavery was abolished in England even earlier. The wikipedia lists a court case in 1500s, where an Englishman brought a slave home from abroad (Russia in this case).

Russia abolished slavery in 1723, but even before 1723 people enslaved by Russians were manumitted and became serfs if they converted to Orthodox faith. After the Great Nordic War there were a few court cases in Finland where former slaves returning home were prosecuted for accepting the Orthodox baptism. Others were not so lucky, Russians sold them to Turkish or Persian markets. A Swedish explorer wrote about his travels in Persia in 1730s. He met slaves who had been captured from Sweden during the Great Nordic War.