I've gone over this before here and here, so let me just summarize. Some people think 1) Jesus Christ is mythological rather than historical, and their primary evidence of this is that 2) there are parallels of Jesus in world mythology. Some take this the further step of arguing that 3) Jesus is completely mythological and thus completely unhistorical; that is, no such person as Jesus of Nazareth ever existed. I'll deal with these in reverse order. In the following, by "scholars" I mean "scholars of the relevant disciplines", i.e. historical Jesus scholars: people with PhDs in ancient history or New Testament history or something similar. I'm sure there are experts in pharmacology or library science who have different views than the scholars I'm referring to, but this is irrelevant since their area of expertise has no bearing on the subject in question. To think otherwise would be to commit the fallacy of irrelevant authority.
3) No scholar thinks it even remotely possible that Jesus may not have existed. Those that do mention such claims explicitly put them on the same intellectual level as Holocaust denial, Moon landing hoax claims, and other conspiracy theories. Indeed, scholars maintain that certain events regarding Jesus are historically certain, and he would obviously have had to exist in order for these events to have taken place. So, for example, Jesus' crucifixion is considered by scholars to be one of the central events in human history; you can't deny it without having to deny most of ancient history in order to be consistent, and it would render subsequent historical development virtually inexplicable. N. T. Wright, the most prestigious contemporary scholar, wrote in The Resurrection of the Son of God that this is true of the empty tomb and post-mortem appearances of Jesus as well: "I regard this conclusion as coming in the same sort of category of historical probability so high as to be historically certain, as the death of Augustus in AD 14 or the fall of Jerusalem in AD 70". Similarly, William Lane Craig has called Jesus' post-mortem appearances "a fact that is almost universally acknowledged by New Testament scholars today".
2) The claim that there are parallels to Jesus in world mythology was only ever held by a minority of scholars, and has been completely rejected by scholars for nearly a century. The parallels in question were conceived so broadly that virtually anything would fit. As such, they were completely contrived. There are, of course, some authors who argue for these parallels even today, but they are not scholars. Joseph Campbell comes to mind: he wrote extensively about mythology and how the Christian myths had many antecedents (except the antecedents were far superior to the Christian version). But Campbell didn't have a PhD, he had a Master's degree in French literature. That's certainly very valuable and a noteworthy accomplishment, but it doesn't qualify him to be considered a historical Jesus scholar. I have a couple of Master's degrees in Philosophy; that doesn't qualify me to be considered a scholar of solid state physics. At any rate, many universities have "The Bible as Literature" courses which are essentially stages to advocate the parallels between Jesus and mythology. But again, these courses are not taught by historical Jesus scholars, they are taught by people with degrees in unrelated disciplines. I find this unfortunate.
1) The idea that the gospels are mythological survived a few decades longer within scholarly circles than did the idea that there are mythological parallels to Jesus. Rudolf Bultmann advocated the view that when the gospels are "demythologized", very little of Jesus could be known beyond the fact that he existed and was killed by crucifixion. Bultmann lived to the 1970s, but his views were rejected by the 1950s with the initiation of the Second Quest for the historical Jesus (we are currently in the midst of the Third Quest). But there is a much more obvious problem with the claim that the gospels are mythological. Mythology is, at least partially, a literary genre, a style of writing. But I'm unaware of any scholar, ever, who argued that the gospels are written in the genre of mythology. Rather, those who claimed they were mythological argued that what the gospels record could not be historical, and so must be mythological, regardless of the genre in which they were written. This point is easily demonstrated: simply read some actual myths -- not modern accounts of myths, but the actual myths themselves -- side by side with the gospels. It's obvious that they don't belong to the same genre, the same type of writing. Thus, James D. G. Dunn argued in the entry for "Myth" in Dictionary of Jesus and the Gospels that the entry wasn't really necessary: "Myth is a term of at least doubtful relevance to the study of Jesus and the Gospels". The genre of the gospels has been a matter of dispute for the last couple of hundred years, although most scholars would have said that they are written as historical writings. But in the last 20-30 years there has been an incredible revolution within historical Jesus studies to the effect that most scholars today consider the gospels to have been written in the literary genre of ancient biography. Of course, this doesn't speak to their reliability in matters of detail, but it certainly makes it difficult to claim they don't have a solid historical core at all.
(cross-posted at Quodlibeta)