The powerful blast was felt across Cairo and black smoke could be seen rising over the city centre.
A second explosion was later heard in the city's Dokki district.
The blasts come on the eve of the third anniversary of the 2011 uprising that led to the removal of the country's decades-long ruler Hosni Mubarak.
No group has said it was behind the attacks.
Tense time
Officials said a booby-trapped car exploded outside Cairo's security directorate.
Gunfire was reportedly heard soon after the blast.
More than 30 ambulances raced to the scene. Most of the wounded are now in hospital, the health ministry says.
State television showed footage of extensive damage to the large concrete building with debris strewn across the street.
The nearby Islamic museum and the National Archives building were also damaged.
The BBC's Orla Guerin, reporting from Cairo, says the security directorate is a very significant target and should have been one of the best protected buildings in the city.
The attack comes at a tense time, with security forces already on heightened alert ahead of the anniversary of the start of the uprising against Mubarak, she adds.
The banned Muslim Brotherhood and its partners are planning demonstrations after Friday prayers across the country.
They are the latest in a series of regular demonstrations against the overthrow in July of Islamist President Mohammed Morsi.
Brotherhood supporters are also angry about recent moves to introduce a new constitution.
BBC
