President Barack Obama said that with the deal, “every pathway to a nuclear weapon is cut off” for Iran.
Iranian President Hassan Rouhani said it opened a “new chapter” in Iran’s relations with the world.
Negotiations between Iran and six world powers began in 2006.
The so-called P5+1 want Iran to scale back its sensitive nuclear activities to ensure that it cannot build a nuclear weapon.
Iran, which wants crippling international sanctions lifted, has always insisted that its nuclear work is peaceful.
In a televised address, President Barack Obama said the deal would make the world “safer and more secure”, and provided for a rigorous verification regime.
“This deal is not built on trust – it is built on verification,” he said.
Immediately afterwards, Hassan Rouhani gave his own televised address, in which he said the prayers of Iranians had “come true”.
He said the deal would lead to the removal of all sanctions, adding: “The sanctions regime was never successful, but at the same time it had affected people’s lives.”
After 12 years, world powers had finally “recognized the nuclear activities of Iran”, he said.
Both Hassan Rouhani and Iranian Foreign Minister Mohammad Javad Zarif referred to the dispute over Iran’s nuclear program as an “unnecessary crisis”.
Earlier, EU foreign policy chief Federica Mogherini said the agreement was “a sign of hope for the entire world”.
“It is a decision that can open the way to a new chapter in international relations,” she said.
Javad Zarif said the deal was “not perfect for anybody”, but that it was the “best achievement possible that could be reached”.
President barack Obama, who is trying to persuade a skeptical US Congress of the benefits, said it would oblige Iran to:
Sanctions relief would be gradual, Barack Obama said, with an arms embargo remaining in place for five years and an embargo on missiles for eight years.
Separately, the IAEA and Iran said they had signed a roadmap to resolve outstanding issues.
IAEA head Yukiya Amano told reporters in Vienna, Austria, that his organization had signed a roadmap “for the clarification of past and present outstanding issues regarding Iran’s nuclear program”.
He called the agreement a “significant step forward”, saying it would allow the agency to “make an assessment of issues relating to possible military dimensions to Iran’s nuclear program by the end of 2015”.
There has been stiff resistance to a deal from conservatives both in Iran and the US. The US Congress has 60 days in which to consider the deal, though Barack Obama said he would veto any attempt to block it.
Israel’s government has also warned against an agreement.
PM Benjamin Netanyahu said it was a “historic mistake” that would provide Iran with “hundreds of billions of dollars with which it can fuel its terror machine and its expansion and aggression throughout the Middle East and across the globe”.
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