Two car bombs exploded near the labour and interior ministries and a suicide attacker driving a car struck a police patrol in Dora, in southern Baghdad, causing 15 of the deaths, an interior ministry official said.
The first explosion in central Baghdad was heard at 10.25 am (0725 GMT) with a second blast within seconds and a third one minute later.
Sporadic gunfire then sounded and the sirens of emergency vehicles were also heard, with helicopters taking to the skies soon afterwards.
The interior ministry official said 12 of those killed by the suicide attacker in Dora were students at a nearby technical college. The remaining three victims were policemen working at the checkpoint.
Violence across Iraq dropped dramatically last month, with the fewest deaths in attacks since the US-led invasion of 2003. Official figures showed a total of 122 people were killed in November.
However the Baghdad government and the US military have warned of a rise in attacks in the run up to the election, which is expected to take place in February.
The threat of political violence linked to the poll is a major concern after bloody attacks in Baghdad in August and October that killed more than 250 people.
The attacks, including truck bombings outside the finance, foreign and justice ministries, punctured confidence in the Iraqi security forces.
"We believe that there will be an attempt to conduct more attacks between now and the election,'' General Ray Odierno, the top US commander in Iraq, said in November.
Tuesday's bombings came two days after the war-torn country's parliament passed a law governing the election, which will be the second national ballot since the 2003 US-led invasion that toppled Saddam Hussein from power.
The United Nations on Monday urged Iraq to announce ``as soon as possible'' the date for the vote after more than two months of delays.
The presidency council, made up of President Jalal Talabani and his two deputies, now has to announce a date for the ballot.
The United Nations Assistance Mission for Iraq (UNAMI) in a statement said that February 27 was ``feasible'' and it stood ready to give all possible help to bring this about.
US diplomats, most notably Christopher Hill, Washington's ambassador to Baghdad, had pushed MPs to pass the law, seeking to avoid delays to the planned pullout of tens of thousands of American troops in 2010.
The United States has 115,000 soldiers in Iraq, but that figure will drop to 50,000 next year as all of its combat troops are pulled out before a complete withdrawal by the end of 2011.