I don't think that anybody denies that Jesus existed as a PERSON (for the most part anyway, never assume...).
I'm not convinced. I have yet to see a non-Biblical source that shows Jesus's existance, and while I've heard they exist no one has ever actually shown me them. Plus, Jesus's story and acts are simmilar to earliar myths. The fraction of the story which seems to be utterly fictional leads me to believe that he very well may have not existed at all. However, this is all just nit-picking, as it doesn't really matter to me if he was a real person or not.
There are a few sources but they're written quite a while after when his death was supposed to occur. Perhaps most commonly cited are the Jewish-scholar Josephus's accounts, both the Testimonium Flavianum and the James passage. There is also a mention by the Roman historian Tacitus with regards to a fire Nero blamed on the Christians. All of them have issues however, and I do not put stock in the Josephus versions.
The Testimonium for instance has the following recorded:
Now there was about this time Jesus, a wise man, if it be lawful to call him a man; for he was a doer of wonderful works, a teacher of such men as receive the truth with pleasure. He drew over to him both many of the Jews and many of the Gentiles. He was [the] Christ. And when Pilate, at the suggestion of the principal men amongst us, had condemned him to the cross, those that loved him at the first did not forsake him; for he appeared to them alive again the third day; as the divine prophets had foretold these and ten thousand other wonderful things concerning him. And the tribe of Christians, so named from him, are not extinct at this day.
There are some huge problems with this text, namely because Josephus remained Jewish and didn't convert to Christianity despite apparently acknowledging a miracle, the fulfillment of prophecy, and in general to ridiculous falling over to praise Jesus. Similarly there is also the usage of Christ. Christ isn't a last name, it's a title for the Anointed One. The Messiah. This peculiarity is found in all 3 mentioned texts.
Most scholars think that the Testimonium is only partially interpolated and that the true text was more like this, although there is still disagreement in many directions:
About this time there lived Jesus, a wise man...For he was one who performed paradoxical deeds and was the teacher of such people as accept the truth gladly. He won over many Jews [and many Greeks?]. He was [called] the Christ. When Pilate, upon hearing him accused by men of the highest standing among us, had condemned him to be crucified, those who had in the first place come to love him did not give up their affection for him...And the tribe of the Christians, so called after him, has still to this day not disappeared.
Next is the other Josephus account. Supposedly describing James the brother of Jesus.
And now Caesar, upon hearing the death of Festus, sent Albinus into Judea, as procurator. But the king deprived Joseph of the high priesthood, and bestowed the succession to that dignity on the son of Ananus, who was also himself called Ananus. Now the report goes that this eldest Ananus proved a most fortunate man; for he had five sons who had all performed the office of a high priest to God, and who had himself enjoyed that dignity a long time formerly, which had never happened to any other of our high priests. But this younger Ananus, who, as we have told you already, took the high priesthood, was a bold man in his temper, and very insolent; he was also of the sect of the Sadducees, who are very rigid in judging offenders, above all the rest of the Jews, as we have already observed; when, therefore, Ananus was of this disposition, he thought he had now a proper opportunity. Festus was now dead, and Albinus was but upon the road; so he assembled the sanhedrin of judges, and brought before them the brother of Jesus, who was called Christ, whose name was James, and some others; and when he had formed an accusation against them as breakers of the law, he delivered them to be stoned: but as for those who seemed the most equitable of the citizens, and such as were the most uneasy at the breach of the laws, they disliked what was done; they also sent to the king, desiring him to send to Ananus that he should act so no more, for that what he had already done was not to be justified; nay, some of them went also to meet Albinus, as he was upon his journey from Alexandria, and informed him that it was not lawful for Ananus to assemble a sanhedrin without his consent. Whereupon Albinus complied with what they said, and wrote in anger to Ananus, and threatened that he would bring him to punishment for what he had done; on which king Agrippa took the high priesthood from him, when he had ruled but three months, and made Jesus, the son of Damneus, high priest
It is the bolded part which causes people to believe this text refers to the Christian Jesus. However, I'm of the mind that that small bit was an interpolation since the entire piece flows together better when Ananus's punishment was to be replaced by the brother of the man he just had stoned to death.
Finally there is the Roman historian Tacitus. He is reporting on a fire and Nero's reaction to blame the Christians for the deed.
Consequently, to get rid of the report, Nero fastened the guilt and inflicted the most exquisite punishments on a class hated for their disgraceful acts, called Chrestians by the populace. Christ, from whom the name had its origin, suffered the extreme penalty during the reign of Tiberius at the hands of one of our procurators, Pontius Pilatus, and a most mischievous superstition, thus checked for the moment, again broke out not only in Judaea, the first source of the evil, but even in Rome, where all things hideous and shameful from every part of the world find their centre and become popular. Accordingly, an arrest was first made of all who pleaded guilty; then, upon their information, an immense multitude was convicted, not so much of the crime of firing the city, as of hatred against mankind. Mockery of every sort was added to their deaths. Covered with the skins of beasts, they were torn by dogs and perished, or were nailed to crosses, or were doomed to the flames and burnt, to serve as a nightly illumination, when daylight had expired.
Besides the strange reference to Christ as a name, this is the account that most convinces me that a Christian Jesus (of some sort) did exist in reality.