Historical theologian here. We have no concrete, substantial evidence for the existence of Jesus beyond the tradition that followed in the late 1st century and forward. In my studies, the most convincing evidence for the existence of Jesus are the writings of Flavius Josephus, who mentions Jesus a couple of times. I say these are the most credible because Josephus was a historian and had no real motivation to propagate the story of Jesus besides telling his story accurately. At the very least, his writings show that by the end of the 1st century, rumors were flying around the Jewish/Roman world about this guy Jesus of Nazareth.

In terms of historical artifacts, the James Ossuary is pretty compelling. An Ossuary is a vessel that contains the bones of the dead, and this one in particular is labeled, "James, son of Joseph, Brother of Jesus." That's pretty much all we got there, and it's not really that credible.

Having that said, biblical scholarship is convinced that Jesus of Nazareth was a historical person. He led a sociopolitical/spiritual revolution during a time of various Jewish political uprisings, all coming to a head with the Jewish Wars, which occurred shortly after the time of Jesus. The gospel accounts, which were written by different people in different cities, give a proper context for Jesus' life and setting in ancient Palestine, and we are able to assume the years Jesus lived based on existing historical records of the sociopolitical structure in that part of the world. We can also infer a lot about Jesus' life/social status by the gospel texts (though we often debate this topic).

Let's not forget that the guy led a group of people that established some community-based followings in various cities within 50 years of his death and all four gospel accounts were written by 120 CE (besides later additions here and there). The gospels of Matthew and Luke seem to have been written using a similar source (named "Q" by scholars), which would have been written even closer to the projected dates of Jesus' life. The Acts of the Apostles and the Pauline Epistles also sort of corroborate the same story, especially involving the Council of Jerusalem (though they give different perspectives). The particular trajectory of Christianity before the 3rd century CE shows that there is some drive behind the formation of the Christian community, and it is not even close to accurate to summarize early Christianity as a continuation of Judaism.

While we definitely have more concrete evidence of dudes like Julius Caesar, it's not outlandish to think of Jesus of Nazareth as a historical figure. We definitely have more evidence of his life than the life of, say, Hermes Trismagistus, Plato, or Philo Judaeus.

At the end of the day, the Romans recorded what mattered to them - Roman people, politics, and affairs. The Jews were a fringe group that were around in the Roman world, but were definitely a minority population, so there doesn't seem to be much interest for the Romans to historically record some renegade Jew that they crucified.

edit: I mentioned the Shroud of Turin. I shouldn'ta done that. edit2: Mark didn't draw from Source Q. My bad.

Follow up question. Is there any historical accounts that are critical of anything Jesus said or did?