In a broad sense, what are the reasons for the history-spanning persecution of Jews?

"Oh my God, they killed Jesus ! (The bastards !)"

In medieval Europe, Jews were unloved for several reasons, the two main of which being:

• The accusation of having "killed Jesus". This had some strong effect in Celtic and Germanic people: justice, at that time and for these people, was both collective and personal. When someone committed a crime, the whole clan/family of the victim was entitled to obtain reparation, and the whole clan/family of the criminal was held responsible. The actual business was conducted between the two clans with no intervening authority (though the local chief or king might have enforced a "price list" for offences, the transaction, be it financial compensation or vengeance, was carried out directly).

  It has been reported that Clovis I, during the few hours of teaching prior to his baptism, had exclaimed that if he had been present with his troops at Jerusalem at the time of the crucifixion, he would have "avenged Jesus". The anecdote is probably fake (all fun anecdotes are), but it illustrates the way people thought at that time. Jews were collectively held liable for the capital offence of deicide.

• Money. Christians were forbidden to "make money", i.e. to lend money with interest. This prohibition was enforced in Catholic areas for a long time, up to about 1830. Jews had no such restriction (well, formally, usury was forbidden, but not in practice). Thus, some Jews played the role of bankers, and became rich, and were creditors to many people. Nobody likes his creditors. This elicited a great deal of mistrust (they were juggling money in un-Christian ways) and jealousy, and then plain hatred. Many anti-semitic stereotypes (Jews portrayed as untrustworthy, avaricious, rich, shifty and treacherous) can be traced back to usury and banking.

  Interestingly, this reason also applies to Islamic countries.

A necessary precondition for any group of people to be persecuted is to be a cohesive, recognizable group. Jews have, throughout history, remarkably maintained specific cultural elements and retained their integrity through generalized endogamy (this is what makes Jews a people, and not only a religion); and yet they were very successful in their dispersion and survival. Many other religiously cohesive groups have suffered persecution (e.g. Zoroastrianism) but did not achieved a universal specific term because the persecutions worked.

One may note that anti-semitism is a worldwide phenomenon only insofar as Christianity and Islam are worldwide phenomena. For instance, China has no history of anti-semitism prior to its recent "westernization" (when Jews were persecuted, it was because they were a "foreign influence", and not specifically targeted at Judaism).

Kudos to @thomasPornin for a decent outline, but I think the answer is broader than that. As I said above, I think the answer is that the Jews do not merit the abuse they've suffered, but I think it may be helpful to outline a few more answers.

• The alien among us. As Mr. Pornin says, Jews have traditionally followed customs that set themselves apart from their neighbors. I will not try to explain why, but I will observe that humanity has a visceral hatred for people who refuse to conform to local norms. Because Jews obstinately continued to be Jews rather than fully assimilating, they have been subject to abuse on the grounds that anything different from us must be brought into conformance with us. If they don't assimilate, then they must think they are better than we are, and the only response to such behavior is to destroy them. @lucian indicates that there may be a problem with this statement, but neglects to specify what the tension is. I will assume that the tension is between Jewish tradition distinguishing Jews from cultural norms, and a desire for cultural conformity. That's a fascinating tension, but one that I don't think can be addressed in a SE answer; it requires a longer treatment that is respectful to both sides of the tension.

• Aliens cause misery - the above problem becomes worse if the community has any bad luck. Humans blame bad luck on those who are different, and on those who cannot retaliate. From witch burning to pogrom, it is obvious to humanity that the cause of all misfortune is aliens living among us.

• Poles live in Poland, Germans in Germany, English people live in England.... I still hear this one today; an acquaintance of mine said this year with a completely straight face that it was utterly unreasonable and wrong that Jews and Gypsy's [his term] argued for toleration. Everyone should just live comfortably among their community and the fact that Jews and the Romany don't have a country make them annoying, and that in order to stop annoying him they should stop being Jews and Rom. Obviously his sense of order in the universe is far more important than their identity. This fell into the category of idiocy that was too large for me to confront that day, but it is another source of hostility. (We will ignore concepts such as Welshmen living in the United Kingdom or Cornishmen living in Britain, or even 500 years of history of Englishmen of African descent. We'll also ignore the notion that "Germany" is a modern concept. ) This level of stupid is not going to be convinced by logic or reason. (this is also the ostensible reason why National Socialism/Fascism has such difficulty dealing with Jews; the underlying ideology says that individuality should be suppressed in favor of a national identity, and cannot deal with internal diversity. Obligatory disclaimer - I'm trying to explain an ideology that is stupid beyond my ability to comprehend; please don't assume that I agree with the thought or the consequences).

As I said, the reasons arise from sociology, psychology, and the fundamental hopeless stupidity of h*m* sapiens. Although it is a problem with historical consequences, it cannot be understood through historical analysis.

The first answer to your question (and the only responsible answer is "nothing; the Jews don't deserve the hatred). The second level is from that noted philosopher Kay, "A person is smart. People are dumb, panicky dangerous animals . . . . " The third answer at a more detailed level is book length, and I suspect, still inadequate.