The blast in central Baghdad took place as a group of policemen were distributing relief supplies to Iraqis who had been driven from their homes during the sectarian bloodshed that followed the 2003 US-led invasion.
Iraqi police said at least five children were among the dead.
"It is a suicide bomber. Obviously that has the fingerprints of al-Qaeda," Major-General Qassim Moussawi, a Baghdad security spokesman, said.
A medic at the nearby Ibn al-Nafis hospital said that five children and a woman were confirmed dead at the hospital.
|Christopher Hill, right, takes up his new posting amid increasing violence in Iraq [Reuters]
The second attack occurred near Muqdadiya, 80km from Baghdad, in the northeastern province of Diyala.
The suicide bomber appeared to have targeted a group of Iranian pilgrims in a restaurant.
Most of the 64 dead are believed to be Iranians visiting Shia Muslim religious sites in Iraq, police said. More than 60 others were wounded.
The attacks were the latest in a series of bombings that have raised fears of a surge in violence as the US military plans for a full withdrawal by the end of 2011.
Security has improved measurably in Iraq over the past two years, but attacks targeting the police are still common in some parts of the country, including the capital.
In another development, Abu Omar al-Baghdadi, the purported leader of the Islamic State of Iraq, an al-Qaeda-linked group, was captured.
Iraqi state television said al-Baghdadi was arrested in eastern Baghdad.
Security experts have previously speculated that al-Baghdadi was a character invented by some extremist groups rather than a real person.
The group is one of a handful of Sunni Arab groups blamed for suicide bombings and other attacks in the northern city of Mosul and other parts of Iraq.