A car bomb exploded near a restaurant and shopping area in the central district of Karrada late on Saturday.
The street was busy with shoppers after sundown in the holy month of Ramadan, which ends this week.
A second bomb exploded later in a predominantly Shia area north of the capital, killing another five people.
The bombings, among the deadliest in Iraq this year, come a week after Iraqi security forces recaptured the city of Falluja from Islamic State (IS) militants.
Authorities say the city was used as a launching pad for attacks on Baghdad by IS.
The jihadist group on Saturday claimed responsibility for the car bomb in Karrada in a statement distributed online by supporters of the hardline Sunni group.
There are reports the source of the blast was a refrigerator van packed with explosives.
Many of those killed in the attack, close to midnight, were children, Associated Press reported.
It caused a huge fire on the main street. Several buildings were badly damaged and were still on fire early on Sunday.
Prime Minister Haider al-Abadi visited the scene in the morning, and was met by angry crowds.
Video posted online appeared to show a convoy, reported to be Mr Abadi's, being pelted with stones.
IS still controls large swathes of territory in the country's north and west, including Mosul, Iraq's second largest city.
But the group has been under pressure in Iraq and in neighbouring Syria, where it has been targeted by government forces and US-backed rebels.
