In the fourth century Christianity spread far and wide across the Roman Empire in a way that it had not in prior centuries. Of course, one of the biggest reasons for this was Constantine's conversion to Christianity in 312 CE. After his conversion, Constantine ordered an end to the persecution of Christians and Christianity began to benefit from the patronage of the empire. Under these favorable circumstances, Christian festivals started to eclipse or replace traditional Roman ones. Christmas, for instance, is celebrated in December 25 because the Roman Church wanted to compete with the Roman celebration of the winter solstice, also celebrated on December 25. In any case, "pagan" holidays were stripped of their protected status in 389 CE. In 391 CE, the emperor Theodosius ordered the cessation of sacrifices and the closing of pagan temples. The following century witnessed the destruction of pagan temples. The Parthenon in Athens was famously converted to a church during the fifth century and many other temples shared the same fate. It is clear that in the course of about 100 years the Roman policy of persecuting Christians was reversed to a policy of patronage towards Christians. By 420 CE the Christian ascetic, Isidore of Pelusium, felt confident saying that "the religion of the Greeks ... has vanished from the Earth."

That may have been the case in the Eastern Roman Empire, in the West paganism lingered on. In 408/9, while Rome was besieged by the Visigoths, the prefect of Rome consulted Etruscan diviners and carried out pagan rituals on the steps of the senate. In the 490s, the bishop of Rome, criticized Christians for partaking in the Lupercalia festival. By the sixth century, however, the practice of Roman paganism in the Western Roman Empire was negligible. Other pagans, such as the Anglo-Saxons brought pagan beliefs to regions formerly under Roman rule. Bishop Martin of Braga (died 579) spoke of preaching against paganism in western Spain. By time of Martin's death, however, the era of Roman paganism was at an end.

Sources:

Price, Simon, and Peter Thonemann. The birth of classical Europe a history from Troy to Augustine. London: Penguin, 2011.

Wickham, Chris. The inheritance of Rome: a history of Europe from 400 to 1000. London: Penguin, 2010.

Note: This is not my area of expertise and I'd be thrilled if anyone cared to elaborate/correct any of the points in this post.

I have a follow-up question that might be something of a request for a definition of "Roman religion." I was at a bar's trivia night and they suggested that Constantine's conversion replaced the Roman religion, as headed by either Sol Invictus or Mithras (they accepted both as the "leading sun god"). Were there official designations or proclamations of the Roman pantheon? Both Sol and Mithras seem to suggest a move beyond the Greco-Roman Olympians (Zeus/Jove, et al.). What made the religious practices "Roman"? Emperor proclamations? Cult Worship?