Daesh, a militant group, claimed responsibility for both the attacks in the Shia majority nation, media reports said.
Suicide bombers and gunmen attacked the parliament building and the Mausoleum of Ayatollah Khomeini in the Iranian capital.
IRIB says that one of four attackers exploded a suicide vest inside the parliament building, though other local news agencies said the explosion may have been caused by grenades thrown by the attackers.
Twelve people were killed and more than 40 injured in the attacks on Tehran's parliament complex and the shrine of revolutionary founder Ayatollah Ruhollah Khomeini. Security forces arrested a "terrorist team" in the planning stages of a third attack, according to the Intelligence Ministry.
It also said a security guard was killed and four people wounded in the shrine attack. The accusations come during heightened tensions between the two countries after Saudi Arabia severed ties with Qatar on the grounds that it supports terrorism alongside Iran.
Officials said the gun attack on the parliament building in the capital, Tehran, was still under way.
"I would like to condemn the attack that took place on Iranian Parliament today and on the mausoleum of Imam Khomeini".
The Assembly expressed its condolences, sympathy and support to the Iranian people on this awful crime, pointing out that Syria and Iran are in the same trench of confronting the Takfiri terrorist organizations with the aim of achieving victory.
State Department spokeswoman Heather Nauert said the USA expressed condolences to the victims and their families.
India, Spain sign 7 agreements, including on civil aviation
Two agreements were also signed on transfer of sentenced persons and waiving of visas for diplomatic passport holders. Discussions on climate change and the possibilities of including India in the G 20 contact also took place.
But Trump also says the USA grieves and prays for innocent victims and for the Iranian people.
At least seven people were killed and 35 others injured in the twin assaults, the semi-official Fars agency reported.
Therefore, he noted, it is inevitable that sooner or later their groups are attacked by security forces everywhere around the world, so they decide to carry out suicide attacks before they are caught. If confirmed, they would be the first assaults by the hardline Sunni Muslim group inside the Shi'ite Muslim country.
Shiite Iran has been singled out as a target by Sunni jihadists, including the Islamic State group, but has largely escaped attacks within its urban centres.
The attackers took control of the parliament and closed all the gates, preventing parliament members from exiting the building.
Iran, the predominant Shiite power, has been helping both Iraq and President Bashar Al-Assad's regime in Syria to battle ISIS.
Iran's supreme leader, Ayatollah Ali Khamenei, dismissed the attacks as mere "fireworks" that would not weaken the country's fight against terrorism, state TV reported.
The attackers managed to get inside the building and engaged in a shootout with the security forces.
The timing of the attacks, shortly after a presidential election in Iran, may indicate that the perpetrators want to cast doubt on the decision of the Iranian people to give President Hassan Rouhani a second term, Middle East expert Catherine Shakdam told RT.
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