As a heads up, historical Jesus is a FAQ on the /r/AskHistorians wiki.

I don't have much to add, so I'm not sure if this will qualify as a top-level comment, but I hope this answers your question, as these answers are all well-written and sourced. Additionally, a book that is often recommended on this topic is Did Jesus Exist?: The Historical Argument for Jesus of Nazareth by Bart Ehrman.

You've already been referred to the FAQ, but one can extend the discussion by considering which standard of evidence we're using: "preponderance of evidence" or "beyond reasonable doubt".

The more or less credible early sources for Jesus' existence are pretty scarce. They basically amount to Paul's epistles, which are valuable because although Paul never saw Jesus, he talked to the people who likely did. Or at least that seems to be the most plausible explanation of his writings. He even writes about meeting James, "brother of the Lord", which is most plausibly interpreted as meeting Jesus' actual brother (and yet the interpretation that this is just a designation of him as a Christian cannot be excluded).

The narrative gospels are at least suspicious as historical sources, because they're heavily mythologized anonymous propaganda written decades later. Moreover, they can most likely be reduced to "Mark", since it is undoubted that both "Matthew" and "Luke" plagiarized from "Mark", and the recent scholarship also shows that "John", the latest and least historically credible gospel of the four canonical ones, is not independent from the synoptic gospels, contrary to what was thought before.

The hypothetical source "Q" on which the shared material of "Luke" and "Matthew" is allegedly based is just that - a hypothesis. Capable mainstream scholars like Mark Goodacre reject this hypothesis in favor of "Luke" using "Matthew". While still widely accepted, this hypothesis is hardly beyond the reasonable doubt.

So it cannot be argued beyond reasonable doubt that the gospels contain traces of numerous independent sources. Material that is contained in some gospel but not in the others doesn't automatically constitute some unknown independent tradition (it should be shown that it's not this particular author's invention for that claim to be made). This is the most problematic part of Ehrman's book about the evidence for Jesus' historicity - he simply assumes independent traditions where he sees new material. So far only one has been shown beyond reasonable doubt: the Markan one.

Earliest non-Christian sources mentioning Jesus are not independent from the Christian ones, so are not of much help to this issue either (so it is not even necessary to discuss e.g. whether Josephus ever mentioned Jesus - although yes, both of "his" references cannot be shown to be authentic beyond reasonable doubt; one was directly falsified, the other is not mentioned by the authors who should have mentioned it and thus could be an inadvertent scribal inclusion of a marginal note into the text).

So basically we're left with Paul, whose letters are plausibly interpreted as describing him meeting Jesus' disciples and brother; and "Mark", a pretty doubtful historical source.

From that one can conclude that yes, probably a historical itinerant apocalyptic preacher Yehoshua existed, who was then mythologized. This seems to be the most parsimonious explanation of the sum of the evidence. So by the standard of the preponderance of evidence (>50%) Jesus existed. And as far as ancient history is concerned, it's not bad.

But is his existence proven beyond reasonable doubt (IOW is it historically certain, as is sometimes claimed)? The sources are scarce and are either of low quality ("Mark") or contain just enough uncertainty as to require certain interpretive assumptions, however plausible (Paul). I'll leave this door open, but in my personal estimation the state of the evidence doesn't permit a conclusion beyond reasonable doubt in this case.