In extradition matters, a state (called the requesting state) wants to prosecute or punish a person for acts alleged to have occurred in that state.
On the other hand, the Requested State claims that the acts in question did not occur, or at least could not have occurred in the Requested State. Further, the Requested State claims that even if technically an offense was committed in its territory, the act did not meet the legal standards required to constitute a crime.
This issue is, of course, highly controversial. The Requested State has every right to try its citizens for crimes committed in its territory. The Requested State also has the right to refuse to surrender its citizens to the Requested State, even if the acts in question would be recognized as crimes under the laws of that country. However, the Requested State also recognizes the right of the Lawful Requesting State to try its citizens for acts committed in the Requesting State.
Therefore, extradition is possible when a Requested State decides to surrender a person to the United States, provided that the Requested State's constitution, laws, and the Treaty (which is structured as a contract between the two states), permit such extradition.
Many nations have abolished the death penalty, sparing defendants from any possible death sentence for capital crimes. In cases where the death penalty may still apply, many nations will not extradite to a country where the death penalty exists (such as the United States).
The United States conducts extradition relatively infrequently in comparison to other jurisdictions. Only 160 countries extradite to the United States. Of those, maybe 50 or 60 have extradition treaties with the US.
Those treaties, of course, must be negotiated and ultimately approved by both Houses of Congress. The process takes years. In the meantime, federal and state officials can negotiate a process of surrender. That usually means lengthy court proceedings.
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