Mossad

The Mossad is the main Israeli intelligence agency. Their mission is to gather intelligence and carry out covert actions on behalf of the State of Israel. They are also a lightning rod for conspiracy theories since they combine two of their favorite targets, Jews and the secretive world of intelligence gathering.

Unlike some other nations' intelligence agencies, they've been more willing to "play dirty," especially when it comes to assassinations and spying on their friends. Or that's the standard for intelligence agencies and they've just been sloppy.

Things they've done

 * Captured Adolf Eichmann from Argentina. Both the BND (German intelligence agency) and the CIA (who the BND had informed) knew of his whereabouts before Mossad got to him. Newer information points to Fritz Bauer, the German attorney general, who would later become instrumental in getting the going, as having been Mossad's source.
 * Extensively and aggressively committed espionage against allies, including the United States.   Before any Americans get upset, please note that the US also spies extensively on its allies...
 * In South Africa, Mossad used the Israeli airline, El Al, as an intelligence asset to spy, an asset it also uses against other nations. Things are less friendly these days between Israel and South Africa in light of "Israel's extensive military and security cooperation with the apartheid regime ousted in 1994" when Mossad brokered dubiously legal, private arms sales to the white-supremacist South Africa.
 * Retaliated against various terrorist groups involved in the 1972 attack on their Olympic team. In an incident known as the Lillehammer Affair, an innocent Moroccan man living in Norway was mistaken for one of the targets and murdered in front of his pregnant wife, leading to most of the assassination team being arrested. Five were convicted; all were deported to Israel within two years. Israel refuses to admit responsibility, though they compensated the widow.
 * Various assassinations or attempts on high ranking Hamas targets. These have also been undertaken by their sister organization, Shin Bet.
 * Drugged and kidnapped nuclear technician and Israeli nuclear weapons program whistleblower Mordechai Vanunu from Italian soil.
 * Likely assassinated Canadian-born engineer, who as part of his quest to economically launch a satellite by cannon developed a series of colossal ultra-long-range artillery pieces. Israel wanted to purchase these from him, but Bull refused and instead accepted an offer from Saddam Hussein's government.
 * Killed various German scientists who were in the Egyptian chemical and nuclear weapons program, after recruiting famous former SS officer (yep, the one who freed Mussolini from his mountain prison after he was overthrown).

Things they didn't do

 * The Kennedy assassinations.
 * The September 11, 2001 attacks.
 * Use an array of animals to spy on other nations (see )
 * Steal an anti-Israel activist's shoe
 * Nearly every other major world event. That's more CIA territory, but even the CIA doesn't have that much power.