Medical officials have said more than 60 people were killed, but Algeria's Interior Minister, Noureddine Yazid Zerhouni, has only confirmed 22 deaths.
The bombs were the latest in a series of attacks in Algeria this year.
No group has admitted responsibility, but Mr Zerhouni has blamed a militant Islamist group linked to al-Qaeda.
A recently-arrested militant had revealed that the Salafist Group for Preaching and Combat (GSPC) had planned several months ago to attack one of Tuesday's targets, Mr Zerhouni said.
BBC Security Correspondent Frank Gardner says the manner of the bombings and choice of targets suggest the involvement of the group, which is now known as al-Qaeda in the Land of the Islamic Maghreb.
"I would like to condemn it in the strongest terms - it cannot be justified in any circumstances"
Earlier, Algerian Prime Minister Abdelaziz Belkhadem called off a cabinet meeting to visit hospitals where the injured were being treated.
"These are crimes that targeted innocent people. Students and school children were among the victims. Nothing can justify the crime," he said.
The UN Secretary General, Ban Ki-moon, said words could not express his sense of shock, outrage and anger at the attack on the UN mission.
"This was an abjectly cowardly strike against civilian officials serving humanity's highest ideals under the UN banner - base, indecent and unjustifiable by even the most barbarous political standard," he said in a statement.
Map showing bomb blast locations
The UN High Commissioner for Refugees, Antonio Guterres, whose staff were caught in the blast, told the BBC that he had "no doubt that the UN was targeted".
The US government called the attacks an act of "senseless violence" on the innocent by the "enemies of humanity".
Students killed
In the attack near the Constitutional Council, a bus packed with students was passing by when the vehicle containing the bomb exploded at around 0930 (0830 GMT).
Security officials said the bus took the full force of the blast and was ripped apart, killing and injuring many of those on board.
Witness: 'Everything fell'
At the UN offices in Hydra, the United Nations Development Programme (UNDP) building bore the brunt of the blast. A residential building and the UNHCR headquarters across the road were also damaged, witnesses said.
Sophie Haspeslagh, who works for the UNDP, told the BBC that she was in a corridor when the blast occurred.
"Everything shattered. Everything fell. I hid under a piece of furniture so I wouldn't be hit by the debris," she said.
"I was holding my jacket on my face because I couldn't breathe."
Ms Haspeslagh said a large part of the UNDP building was destroyed and it was feared people were trapped inside.
A UN spokeswoman, Marie Okabe, said unconfirmed preliminary information indicated that four UN staff members may have been killed.
"We're still trying to account for 14 people," she told reporters in New York.
However, the Algerian interior minister said no UN personnel had been among the 12 dead bodies so far recovered in Hydra.
'Suicide bomber'
Mr Zerhouni said the explosions had been caused by two car bombs, and that the one at the UN was triggered by a suicide bomber.
Ms Haspeslagh said one of her colleagues had seen a white van drive into the main UN offices then explode.
There have been a series of bomb attacks across Algeria during the past year in which scores of people have died.
In September more than 50 people were killed in suicide attacks - one of them involved a truck packed with explosives being driven into a coast-guard base.
Al-Qaeda link?
Members of the public have recently held rallies in protest at the upsurge in violence.
VIOLENCE IN 2007
Many of the recent blasts have been claimed by members of al-Qaeda's North Africa wing, calling themselves al-Qaeda in the Land of the Islamic Maghreb (AQLIM), including a triple suicide bombing in Algiers in April which killed 33 people.
The militant group was previously known as the GSPC, but changed its name when it reportedly joined forces with al-Qaeda last year.
BBC regional analyst Roger Hardy says it is unclear how far the group really is linked to Osama Bin Laden's organisation, and how far it is merely inspired by it.
What is worrying Western experts and North African governments is the possibility that radical Islamists in the region no longer have a merely local agenda but are linked to a wider web of international networks.
Algeria suffered a brutal and bloody civil war in the 1990s, but in recent years violence had declined.
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