The Islamist group Boko Haram said it carried out the attacks, including one on St. Theresa’s Church in Madalla, near the capital Abuja, that killed 35.
A second explosion shortly after hit a church in the central city of Jos. A policeman died during gunfire.
Three attacks in northern Yobe state left four people dead.
Two hit the town of Damaturu, and a third struck Gadaka. Yobe state has been the epicentre of violence between security forces and Boko Haram militants.
President Goodluck Jonathan, who is a Christian, said the attacks were an “unwarranted affront on our collective safety and freedom”.
The White House condemned what it described as “senseless violence” and pledged to assist Nigeria in bringing those responsible to justice.
The group carried out an August 2011 suicide attack on the UN headquarters in Abuja, in which more than 20 people were killed.
Nearly 70 people have died this week in fighting between Nigerian forces and Boko Haram gunmen in the country’s north-east.
National Emergency Management Agency (NEMA) spokesman Yushau Shuaibu told the BBC that the latest Abuja explosion had happened in the street outside the church.
He said the church – which can hold up to 1,000 people – had been badly affected by the blast.
Witnesses said windows of nearby houses had been shattered by the explosion.
Officials at the local hospital said the condition of many of the injured was serious, and they were seeking help from bigger medical facilities.
Police had trouble controlling the anger of local people.
Reuters reports that thousands of youths have erected roadblocks on the road from the capital to the largely Muslim north, and are being tackled by security forces firing tear gas.
One of the Damaturu explosions was a suicide car bomb attack on a convoy of the State Security Service.
Two explosive devices found in a nearby building were disarmed as military were deployed to the site.
The attack in Jos, in Plateau state, could have even more serious consequences than the attack in Abuja.
The state lies in Nigeria’s so-called Middle Belt, between the mainly Muslim north and Christian south.
More than 1,000 have been killed in religious and ethnic violence in Jos over the past two years and our correspondent says there will be fears that the latest attack could spark wider conflict.
A string of bomb blasts in Jos on Christmas Eve 2010 were claimed by Boko Haram.
President Goodluck Jonathan said after the latest attacks: “I want to reassure all Nigerians that the government will not relent in its determination to bring to justice all the perpetrators.”
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