Both Matthew and Luke place Jesus's birth at Bethlehem, but the two Gospels reach that location and narrate the surrounding events quite differently. Luke frames the birth around a Roman census that brings Joseph and the pregnant Mary from Nazareth to Bethlehem, the ancestral city of David, where Jesus is born and laid in a manger because there was no room at the inn, a phrase whose translation and precise meaning continue to be debated by scholars. Matthew, by contrast, does not mention a census or a journey from Nazareth at all, and appears instead to assume that Joseph and Mary already lived in Bethlehem, only relocating to Nazareth later, after their return from Egypt, because Matthew's narrative requires a specific explanation for how the family came to live there instead.
Matthew places the birth during the reign of Herod the Great, who is independently known to have died in 4 BCE, while Luke ties his account to a census under the Roman governor Quirinius, independently attested but dated by the historian Josephus to 6 CE, a full decade after Herod's death, a discrepancy scholars have not been able to fully reconcile.
Neither Gospel gives an explicit year using a dating system directly comparable to modern chronology, and the widely used calendar era counting years from Jesus's birth, established centuries later by the monk Dionysius Exiguus, is now generally regarded by historians as having miscalculated the actual year by several years in either direction.