Talabani was president from 2005 to 2014 and a key figure in Iraqi Kurdistan, where voters last week overwhelmingly backed independence in a disputed referendum.
An official from Talabani's Patriotic Union of Kurdistan (PUK), which was initially sceptical about the vote before agreeing to it, said he had died in Germany where he was undergoing medical treatment.
A family member said his health had worsened and he been transported to Germany, along with his wife and two children, before the referendum.
Prime Minister Haider al-Abadi, a fierce opponent of the referendum, hailed Talabani for his role in "building a federal Iraq".
"He described Iraq as a bouquet made up of several flowers," he said, referring to the country's different communities.
His political rival and the current leader of Iraqi Kurdistan, Massud Barzani, also paid his respects to Talabani.
"I lost a friend and a brother. I'm glad we fought together," said Barzani, who announced a week of mourning during which Kurdish flags would be flown at half-mast.
'Great void'
German Foreign Minister Sigmar Gabriel, in a statement, paid tribute to Talabani for his "great service to his country at a difficult time".
Iraqi Kurdish lawmaker Zana Said described Talabani as "the only president whose death saddens Arabs, Kurds and all other ethnicities".
"We pray to God that his death will help to bring back good relations between the brothers of Iraq."
Masrur Barzani, a senior member of the Kurdistan Democratic Party (KDP), also lamented his death.
"The disappearance of Uncle Jalal will leave a great void because he was a great political figure in both Kurdistan and Iraq," he said, using his nickname.
Talabani's death, following a decades-old struggle for Kurdish statehood, came after Iraq's Kurds voted 92.7 percent in favour of independence in the September 25 referendum.
The vote, rejected by Baghdad as illegal, has deeply strained ties between the Kurds and central Iraqi authorities, who have cut off international flights to the region and threatened further action.
Talabani was an avuncular politician and a skilled negotiator who spent years building bridges between the country's divided factions, despite his efforts for Kurdish independence.
Uncle Jalal
Born in 1933 in the mountain village of Kalkan, he studied law at Baghdad University and did a stint in the army before joining the KDP of Mullah Mustafa Barzani, father of the current Kurdistan regional president.
Talabani took to the hills in a first uprising against the Iraqi government in 1961 but famously fell out with Barzani, who sued for peace with Baghdad, and joined a KDP splinter faction in 1964.
Eleven years later, he established the PUK after Barzani's forces, abandoned by their Iranian, US and Israeli allies, were routed by Saddam Hussein's army.
He became president in April 2005 after the first post-Saddam election in Iraq and continued in the post until 2014, when he was replaced by the current president, Fuad Masum.
Iraq's head of state plays a largely ceremonial role and is elected by members of parliament.
In his mountainous northern fiefdom of Sulaimaniyah, Talabani was known affectionately as Mam (Uncle) Jalal.
In August 2008, the married father of two underwent successful heart surgery in the United States, then in 2012 he was flown to Germany after suffering a stroke, casting doubt on whether he would return to Iraq.
He did go back in July 2014, with Iraq in crisis after the Islamic State group had taken control of swathes of the country, and was replaced by Masum following a parliamentary election.
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