The Jesus as we know him - the Jesus of Christianity - definitely did not exist. It is possible that there was no historical Jesus at all. No historians of the first century mention Jesus, despite there being authors who write (at length) about Jewish concerns. There are no Roman records that mention Jesus1. Not only all that, but, there are no Christian eye-witnesses of Jesus. All of the Gospels are anonymous and written by friends-of-friends, and none are written in the first person; also, Paul (who authored 13 of the 27 books of the NT) never met Jesus, except in a vision2. They're also written in very competent Greek (the language of later converts), rather than in Hebrew (the language of the original converts, excepting Paul). Early Christians didn't know when Jesus was born (his birthdate wasn't decided for hundreds of years, in 354CE)3 and didn't know where he was buried. People have doubted his existence since the very first century, and, despite the popularity of Christianity, there is a modern resurgence of people who disbelieve in the very existence of Jesus today. The biggest problem facing such unbelievers is accounting for early Christianity. But there are multiple theories as to how Christianity may have arisen without a historical saviour. For example, it is quite possible, given the similarities of Jesus to previous saviour religions and pagan stories about god-men, that the entire story of Jesus is a rewrite, with Jewish undertones, of Roman and pagan myths that were current at the beginning of the first century4, 5, 6.
Caveat: Nothing in the Bible was written by Jesus.
