The first explosion happened in the Ben Aknoun district, near the supreme constitutional court.
That was followed shortly afterwards by a second blast at the United Nations offices in the Hydra neighbourhood
A UN worker caught up in the Hydra attack told the BBC that a large part of the building was destroyed and it is feared people are trapped inside.
Dozens were also wounded in the explosions, officials say.
Students killed
In the attack near the court a bus packed with university students was passing by the vehicle containing the bomb when it exploded.
Security officials say that the bus took the full force of the blast and was ripped apart, killing and injuring many of those on board.
At the UN offices in Hydra it was the UNDP building which bore the brunt of the blast. A residential building and the UNHCR headquarters across the road were also damaged, witnesses say.
Sophie Haspeslagh, who works for the United Nations Development Programme (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."
'Suicide bomber'
Algerian Interior Minister Yazid Zerhouni said the explosions had been caused by two car bombs, and that the UN one was triggered by a suicide bomber.
There has been no immediate claim of responsibility.
Throughout 2007 there have been a series of bomb attacks across Algeria 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.
Many of the recent blasts have been claimed by members of al-Qaeda's North Africa wing, calling themselves al-Qaeda in the Islamic Maghreb, including a triple suicide in Algiers in April which killed 33 people.
The militant group was previously known as the Salafist Group for Preaching and Combat (GSPC) but changed its name when it joined forces with al-Qaeda last year.
The BBC's 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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