Some critics of Christianity claim that there is no evidence that Nazareth existed at the time of Jesus.  This claim is often associated with Jesus Mythicism. This argument has largely been refuted by evidence. The priestly inscription is rather clear, unbiased and impartial evidence that suggests that a town called Nazareth existed and was large enough to take in a priest after the fall of the Temple in 70AD. 
- No one knows where Nazareth was, but it was big enough to be a city that had a synagogue.
- No ancient historian or geographer mentions Nazareth. Nazareth is first noted at the beginning of the 4th century.
- Luke 28-30 tells that Nazareth was on a hill and that the hill had a cliff high enough that a man falling off it would be killed. The recently occupied town now called Nazareth, however, has never occupied the top of a hill. 
- Nazareth does not appear in the Old Testament. The Book of Joshua 19:10-16 – in what it claims is the process of settlement by the tribe of Zebulon in the area – records twelve towns and six villages and yet omits any 'Nazareth' from its list.
- Nazareth does not appear in the volumes of Josephus's writings (even though he provides a detailed list of the cities of Galilee). The Talmud, although it names 63 Galilean towns, knows nothing of Nazareth, nor does early rabbinic literature.
- Oddly, Paul the Apostle knows nothing of 'Nazareth'. Rabbi Solly's epistles (real and fake) mention Jesus 221 times, Nazareth not at all. None of the New Testament epistle writers ever mentions Nazareth or a Jesus of Nazareth even though most of the epistles got written before the gospels. In fact no one mentions Nazareth until the Gospels, where the first one didn't come into existence until about 40 years after the hypothetical death of Jesus. Apologists attempt to dismiss this by claiming that Nazareth existed as an insignificant and easily missed village (how would they know?), thus no one recorded it. However, whenever the Gospels speak of Nazareth, they always refer to it as a city, never a village (Luke 1:26-27 ; Luke 2:3-4 ; Matthew 2:22-23 ; Luke 2:39-40 ), and a historian of that period would surely have noticed a city. (Note the New Testament uses the terms village, town, and city.)
"Your middle claim could be true (some peer reviewed discussions of late seem to concede the possibility that there is no definite evidence of an early 1st-century Nazareth), though there is a difference between not having evidence and the town not being there. Personally, I find it hard to believe the town would suddenly appear and get that name just in time to take in priests after the first Jewish War ... I leave it to the experts to debate the matter. Until there is a consensus against an early 1st century Nazareth, we should be skeptical of claims to the contrary."