Contrary to some circles on the Internet, very few scholars doubt that Jesus existed, preached and led a movement. Scholars' confidence has nothing to do with theology but much to do with historiographic common sense. What movement would make up a recent leader, executed by a Roman governor for treason, and then declare, "We're his followers"? If they wanted to commit suicide, there were simpler ways to do it.
One popular objection is that only Christians wrote anything about Jesus. This objection is neither entirely true nor does it reckon with the nature of ancient sources. It usually comes from people who have not worked much with ancient history. Only a small proportion of information from antiquity survives, yet it is often sufficient.
We recognize that most people write only about what they care about. The only substantive early works about Socrates derive from his followers. The Dead Sea Scrolls extol their community's founder, but no other reports of him survive. The Jewish historian Josephus claims to be a Pharisee, yet never mentions Hillel, who is famous in Pharisees' traditions. Israeli scholar David Flusser correctly observes that it is usually followers who preserve what is most meaningful about their teachers, whether the leaders were Buddha, Muhammad, Mormon leader Joseph Smith or African prophet Simon Kimbangu.
Interestingly, however, once ancient writers had reasons to care about Jesus, they did mention him.
Josephus, the only extant first-century historian focused on Judea, mentions both Jesus and John the Baptist as major prophetic figures, as well as subsequently noting Jesus' brother, James. Later scribes added to the Jesus passage, but the majority of specialists agree on the basic substance of the original, a substance now confirmed by a manuscript that apparently reflects the pre-tampering reading. Josephus describes Jesus as a sage and worker of wonders, and notes that the Roman governor Pilate had him crucified. On the cause of crucifixion Josephus remains discreet, but mass leaders were often executed for sedition -- especially for being potential kings. Perhaps not coincidentally, Jesus' followers also insisted, even after his death, that he was a king. Josephus was not a Christian and does not elaborate, but his summary matches other sources.
Writing even earlier than Josephus, Syrian philosopher Mara bar Sarapion claimed that Jesus was a wise Jewish king. Tacitus later reports on events from 31-34 years after Jesus' ministry, associating Roman Christians with him and noting that he was executed under Pontius Pilate. These and other sources provide only snippets, but they address what these sources cared about. By comparison, Tacitus mentions only in passing a Jewish king on whom Josephus focused (Agrippa I); nor was Tacitus interested even in Judea's Roman governors. Tacitus's mention of Pilate in connection with Jesus' crucifixion is Roman literature's only mention of Pilate (though Pilate appears in Josephus and an inscription).
From Jesus' followers, who were interested, we naturally learn much more. Fifteen to 30 years after Jesus' ministry, Paul wrote much about Jesus, including an encounter that Paul believed he had with the risen Jesus probably within a few years of Jesus' execution. Rightly or wrongly, Paul staked the rest of his life on this experience. Other early Christians also preserved information; some 30-40 years after Jesus' ministry, Mark's Gospel circulated. Luke reports that "many" had already written accounts by the time Luke writes. Luke shares with Matthew some common material that most scholars think is even earlier than Mark. Only a small minority of figures in antiquity had surviving works written about them so soon after their deaths.
What can the first-century Gospels tell us? Certainly at the least they indicate that Jesus was a historical figure. Myths and even legends normally involved characters placed centuries in the distant past. People wrote novels, but not novels claiming that a fictitious character actually lived a generation or two before they wrote. Ancient readers would most likely approach the Gospels as biographies, as a majority of scholars today suggest. Biographies of recent figures were not only about real figures, but they typically preserved much information. One can demonstrate this preservation by simply comparing the works of biographers and historians about then-recent figures, say Tacitus and Suetonius writing about Otho.
What was true of biographies in general could be even more true of biographies about sages. Members of sages' schools in this period typically preserved their masters' teachings, which became foundational for their communities. Memorization and passing on teachings were central. Oral societies were much better at this than most of us in the West today imagine; indeed, even illiterate bards could often recite all of Homer from heart. None of this means that the Gospels preserve Jesus' teaching verbatim, but by normal standards for ancient history, we should assume that at the least many key themes (e.g., God's "kingdom") were preserved. Indeed, many of the eyewitnesses (such as Peter) remained in key leadership positions in the movement's earliest decades.
One significant feature of these first-century Gospels is the amount of material in them that fits a first-century Galilean setting. That setting differs from the Gospel writers' own setting. The Gospel writers updated language to apply it to their own audiences, but they also preserved a vast amount of information. This is merely a sample; specialists devote their lives to the details.
Yet, valuable as examining such historical evidence is, we must return to where we started. Logically, why would Jesus' followers make up a Jesus to live and die for? Why not glorify real founders (as movements normally did)? Why make up a leader and have him executed on a Roman cross? To follow one executed for treason was itself treason. To follow a crucified leader was to court persecution. Some people do give their lives for their beliefs, but for beliefs, not normally for what they know to be fabricated. Jesus' first movement would not have made up his execution or his existence. How much they actually remembered about him is a subject for a future post.