FACTBOX - Ups and downs of Iraq-Iran relations

(Reuters) - Mahmoud Ahmadinejad is visiting Iraq, the first Iranian president to do so since the two neighbours fought a war in the 1980s in which 1 million people were killed.

Tehran's influence over Baghdad has grown since the U.S.-led invasion toppled Iran's long-time enemy Saddam Hussein in 2003. Following are details about relations in the past 30 years:

* IRAN-IRAQ WAR:

-- September 1980 - Iraq accused Iran of shelling Iraqi border towns from territory belonging to Baghdad under the 1975 Algiers accord on the frontier line and Shatt al-Arab waterway. Saddam Hussein tore up the accord and his troops invaded Iran.

-- March 1988 - Iran seized the town of Halabja in northeast Iraq. Tehran said Iraq used chemical weapons to punish inhabitants for not resisting. About 5,000 people were killed.

-- August 1988 - The Middle East's longest armed conflict in modern times ended with a U.N.-sponsored ceasefire. About one million people were killed in a conflict which involved trench warfare, chemical attacks and mass Iranian frontal assaults.

* GULF WAR:

-- August 1990 - Iraq invaded Kuwait. Days later Saddam told Iran he would withdraw from occupied Iranian territory and formally settle the 1980-88 war.

-- January 1991 - Iraq flew more than 140 aircraft to Iran to avoid their destruction before a U.S.-led attack to force Iraqi troops out of Kuwait. Iraq's air force commander said in 2007 he hoped Iran would return some Iraqi warplanes.

* IMPROVING TIES

-- 1998 - Iran's moderate president, Mohammad Khatami, broke a logjam in ties, leading to swapping more prisoners from the 1980-88 war and relaunching pilgrimages by Iranians to holy Shi'ite Muslim sites in Iraq.

-- 2003 - Iran-Iraq ties improve after Saddam was toppled. An Iraqi government led by Shi'ite Muslims came to power. Iran is a Shi'ite nation.

-- The U.S. military has accused Iran of stoking violence in Iraq by funding, training and equipping militias. Iran denies this, blaming the presence of U.S. troops for the violence.

-- More recently Iran may have used its influence with Iraqi Shi'ite militias to help reduce violence as a concession to Washington when Tehran was worried about a U.S. threat to resort to force to deal with a standoff over its nuclear programme.

-- Iraq has also credited Tehran with helping to rein in the Mehdi Army militia of Shi'ite cleric Moqtada al-Sadr.

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