It's not likely an empire wide census happened in the way that the author of the Gospel of Luke described it. Cyrenius, the governor of Syria, we're told by Josephus, conducted a census of Syria and Judea in 6-7 CE, and that's probably the one that Luke is referring to. In fact, Luke specifically says that the census happened "in the days that Quirnius was governor of Syria". (Quirnius is another transliteration of Cyrenius).
Here's Josephus, from Antiquities:
Now Cyrenius, a Roman senator, and one who had gone through other magistracies, and had passed through them till he had been consul, and one who, on other accounts, was of great dignity, came at this time into Syria, with a few others, being sent by Caesar to be a judge of that nation, and to take an account of their substance. Coponius also, a man of the equestrian order, was sent together with him, to have the supreme power over the Jews. Moreover, Cyrenius came himself into Judea, which was now added to the province of Syria, to take an account of their substance, and to dispose of Archelaus's money; but the Jews, although at the beginning they took the report of a taxation heinously, yet did they leave off any further opposition to it, by the persuasion of Joazar, who was the son of Beethus, and high priest; so they, being over-pesuaded by Joazar's words, gave an account of their estates, without any dispute about it.
Note, that the Gospel of Luke doesn't say that Jesus was born during the reign of Herod the Great. It says that John the Baptist (who had apparently been conceived but not yet born when Jesus was conceived), was born "in the time of Herod king of Judea", but that could just as easily refer to Archelaus.