Many Christian apologists claim that the resurrection of Jesus is an established historical fact, and that this proves the truth of Christianity.
The Problem of Miracles
Main Article: Miracles in History One problem with this line of apologetic argument is that it is difficult to establish historically that a miracle has occurred. The reasons for this include practical difficulties such as the many cases of definitely bogus miracle claims as well as the epistemic problem raised by Hume. However, the most basic reason may be that most Christians would demand a very high level of proof before accepting a miracle of another religion. Therefore, we should apply a similar standard to Christian miracles such as the resurrection.
There is very little historical documentation for the resurrection outside the Bible itself. Apologists must therefore claim that the books of the Bible can be established as reliable historical documents.
Main article: The Gospels Today, even many fairly conservative scholars admit that the gospels are not eyewitness accounts but rather anonymous compositions written decades after the alleged events they record. As hearsay, they are almost entirely worthless as proof of an actual miracle.
In I Corinthians 15, Paul makes a briefly lists some post mortem appearances of Jesus, including appearances to himself and the disciples. As evidence for the resurrection, it has an advantage over the gospel reports in that most scholars agree that Paul really wrote the relevant passage from I Corinthians. (A few scholars, most notably Robert M. Price, disagree with this conclusion, however.)
Nevertheless, there are problems with Paul's report. It lacks all the details of the gospel reports which apologists use to show an actual miracle is the only viable explanation for the evidence. Conspicuously absent is any reference to the discovery of Jesus' tomb, a fact that has led some scholars to question the historicity of the tomb story.
Paul says nothing about how he received his information, leaving a real possibility that some of it has been mis-reported.
In public debates, Anthony Flew has pointed out that organizations such as the Society for Psychical Research (of which Flew is a former member) would not accept Paul's report as evidence of ghosts or telepathic projection. The problem is that experience with such reports shows that human memory is much less reliable than most people realize, and even a year after an event is long enough to introduce significant distortions in recollection. Paul's letter to the Corinthians was written not a year after the fact, but two to three decades after.
The evidence for the resurrection is open to a number of naturalistic explanations
"Swoon theory" refers to the hypothesis that Jesus didn't really die on the cross, but rather was taken down alive and recovered in the tomb. It was made famous in the 19th century by Heinrich Paulus, as well as by fictional works that postulated an Essene conspiracy that assisted in the ruse. Today it has few advocates, though Richard Carrier recently published a partial defense of it.  Carrier argued that it was actually the least likely naturalistic explanation, but it still had a chance of occurring of 1 in 6,800. This is sufficient to rule out a miracle, because if every 1 in 6,800 event were declared miraculous, we would have to believe that royal flushes are miraculous.
Keith Parsons has recently argued that recent experience with people who believe themselves to have been abducted by aliens makes the hallucination hypothesis more plausible, and that many standard apologetic objections to the hypothesis would also require us to believe in alien abduction.
Biblical scholar Dale Allison has made a similar argument based on reports of apparitions of the dead. In particular, he notes "examples of collective hallucinations in which people claimed to see the same thing but, when closely interviewed, disagreed on the details, proving they were not, after all, seeing the same thing."
The standard objection to the fraud theory is that the disciples would not have died for a lie. However, documentation of their martyrdoms is weak. The earliest comes at the end of the 2nd century and is only for Peter and Paul. Also, it has been suggested that the disciples may have lied for what they believed was a higher cause.
The resurrection and Jesus mythicism
To some extent, the debate over the resurrection would be moot if it were demonstrated that Jesus never existed. However, some mythicists, notably Richard Carrier, accept that early Christians reported visions of Jesus, and these are explained as hallucinations.
Allison, Dale. Resurrecting Jesus T. & T. Clark Publishers, 2005
Price, Robert M. and Jeffery Jay Lowder (editors). The Empty Tomb Prometheus Books, 2005