When does a state have authority to prosecute crimes committed elsewhere?

Inspired by this thread on a Spanish judge’s recent warrant to arrest three U.S. soldiers accused of killing a Spanish journalist in Baghdad: http://boards.straightdope.com/sdmb/showthread.php?t=340572&page=1&pp=50

The case is based on a Spanish law giving the courts jurisdiction over homicide of a Spanish citizen if the host country fails to prosecute. Some commentators go further and call it an instance of “universal jurisdiction.” Let’s debate that.

Under what circumstances, if any, does a state’s government or its court system have legitimate jurisdiction to prosecute crimes alleged to have been committed outside the state’s territory?

Did the UK courts have proper jurisdiction over Augusto Pinochet?

Did the U.S. have the authority to arrest and prosecute Manuel Noriega?

Did the State of Israel have the right to arrest (or kidnap) Adolf Eichmann in a foreign country without the permission of its government, and then try him for crimes committed in other countries before the State of Israel even existed?

Is the authority of international bodies any more far-reaching? E.g., the trial of Slobodan Milosevic before the UN International Criminal Tribunal for the Former Yugoslavia – was that a legitimate exercise of authority?

Are you talking legally, philosophically or in reality?

In reality, it’s whenever they have the ability to do so and nobody will do anything about it.

Legally and philosophically.

As I recall, Pinochet went to Britain for medical care. While there, the Brits held Pinochet while trying to figure out if they had to extradite him to Spain for an investigation into the deaths / disappearances of Spanish citizens while he was in power (presumably at the hands of the state security forces).

Well, I’ve got a pile of cites towards the end of that thread, on the topic. Near as I can tell, according to “Universal Jurisdiction”, whenever a state feels that someone wasn’t punished for a REALLY BAD thing like murder (And they get to define murder - as it seems killing people as a soldier during warfare is now potentially murder), they have the right, nay, the duty, to try them.

And if another state feels like it after that, they can try them.
And so on, and so on.

Personally, I wonder why an Iranian court hasn’t tried to try George Bush for war crimes. Or Ariel Sharon, either way.

In effect, it seems to be useful either against the powerless, or those no-longer in power and very unpopular.