But who is Abadi? Born in Baghdad in 1952, Abadi was educated at the University of Baghdad and later received a doctorate from the University of Manchester in Britain. He lived in Britain for many years after his family was targeted by Saddam Hussein's Baathist regime. He was trained as an electrical engineer, but he entered politics after the U.S.-led invasion of Iraq in 2003. He became minister of communications in the Iraqi Governing Council in September 2003, then was a key adviser to Maliki in Iraq's first post-invasion elected government. Just weeks ago, he was elected deputy speaker of parliament, and he has been considered a contender for prime minister after the past two elections.
The bigger question, however, is whether Abadi will be able to overcome the challenges confronting Iraq more successfully than Maliki. Like Maliki, he's a Shiite Muslim and is a member of the ruling State of Law coalition. One of the chief criticisms of Maliki was that he entrenched Iraq's sectarian politics, filling the government with Shiite politicians and limiting Sunni and Kurdish power.
Earlier this summer, Abadi gave a striking interview to the Huffington Post's Mehdi Hasan in which he discussed the possibility of Iranian intervention in the fight against the Islamic State, the Sunni extremist group that has taken over vast swaths of Iraq.
"We are waiting for the Americans to give us support," he said in the June interview. "If U.S. air strikes [happen], we don't need Iranian air strikes. If they don't, then we may need Iranian strikes." Abadi has also had differences with Iraq's Kurdish community at points: Last year he warned that a dispute over Iraqi Kurdistan's oil exports could lead to the "disintegration" of the country, and he was criticized by Kurdish politicians during the negotiations over the 2013 budget.
However, Abadi does seem to be aware that the Iraqi government and security forces have made serious mistakes in the current conflict. He told Hasan that the government needs to listen to stories of the "excesses" of the security forces to decide how to respond. And he was clear that Iraq needed to avoid being dragged into the type of war the Islamic State clearly desires.
"We have to be careful not to become involved in a sectarian war," he told Hasan. "Shias are not against Sunnis and Sunnis are not against Shias."
Reidar Visser, an academic expert on Iraqi politics, says that although Abadi comes from the same political faction as Maliki, he enjoys much broader support, especially from Kurds and Sunnis.
Part of this is his more distinguished background. "Many of the elites from the governing council-era consider him one of their own in terms of a prestigious family background, whereas Maliki was seen as more of an upstart from humble origins," Visser notes in an e-mail. "Things like that count in the (old-fashioned and traditional) Iraqi establishment."
Hayder al-Khoei, an associate fellow at Chatham House in London who follows Iraqi politics, says that Abadi has a reputation for skilled diplomacy, and has a much better chance of forming a national-unity government than his predecessor. "Without Maliki it isn't going to be easy," Khoei explains, "[But] with Maliki it will be impossible."
Khoei added that it remains to be seen exactly how Maliki will react to the appointment. "To give you an indication of how bad the situation is, many are now worried about Abadi's physical security," he said.
President Fouad Massoum, a Kurd, seemed confident Monday that Abadi could lead the country. “Now the Iraqi people are in your hands,” he said as he shook Abadi's hand. Now Iraq waits to see whether Maliki will acquiesce.
Rick Noack contributed to this report.