A bomb blast near the city's main bus station killed at least eight people and wounded 28, hitting a street market popular with
Iraqi Shi'ites travelling by road to southern cities.
Four policemen were among those hurt, the interior ministry said.
Later, two more bombs went off in rapid succession, killing 14 more people and wounding 65, according to an interior ministry official.
A thick column of smoke poured into the air above Sadun Street in the heart of Baghdad after the explosions, which damaged a row of shops and a popular coffee shop.
Security sources said the bombers appeared to have been targeting civilians in an area with few homes which is popular with Baghdad's Christian and Kurdish minorities.
The Iraqi capital is in the grip of a sectarian war in which
Sunni insurgents target crowds of mainly Shi'ite civilians in bomb attacks, and Shi'ite death squads carry out kidnappings and reprisal killings.
Iraqi and US security forces have launched a large-scale campaign dubbed Operation Together Forward to isolate flashpoint neighbourhoods and conduct house-to-house weapons searches.
Government security forces and their US advisers carry out nightly raids against suspected death squads or bomb-making cells, and in recent days have reported several apparent successes.
But there is little sign yet of the violence abating, amid warnings that Iraq may be sliding towards all-out sectarian war.