The self-proclaimed Islamic State, or IS, has claimed responsibility for the attack.
It comes amid a fresh push to end Yemen's 17-month war between the Saudi-backed government and the Houthi rebels who took over the capital Sana'a.
As new recruits gathered at a staging area inside their training camp in the city of Aden, the attacker made his move.
Witnesses say they saw a pickup truck suddenly accelerate through the building's gate and explode amid the crowd.
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More than 50 people were killed at the scene, and almost 70 more were wounded, many seriously.
IS has declared one of its suicide bombers carried out the attack in a statement on its Amaq news agency.
"Around 60 are dead in a martyrdom operation by a fighter from IS targeting a recruitment centre in the city of Aden."
The deadly attack in the southern Yemeni port city comes amid a civil war in Yemen.
The internationally recognised government, backed by a Saudi-led coalition, has been fighting Houthi rebels allied with army units loyal to a former president.
But the fighting has allowed al-Qaeda and IS to expand their reach, particularly in the country's south, where the government was forced to flee and set up its base.
There have been attempts at peace talks between the government and the Houthis, but sporadic attacks have stalled negotiations.
Last week, the United States announced a new Gulf-backed initiative, which involves forming a unity government as Houthi rebels withdraw from the capital, Sana'a.
The head of the Houthi-backed ruling council, Saleh Al Sammad, says the council is ready to restart peace talks, but only if the Saudi-led coalition stop attacks on its territories.
"I believe there are serious international intentions to reach peace, and we will do our best to take any chance to stop the aggression and lift the suffering from our Yemeni people."
but Mr Sammad says the Houthis have a right to strike back in response to military attacks.
"We have up to a hundred raids a day. Just yesterday, there were more than 81 raids on the Sheibani area. And we have no planes or the kind of deadly weapons they have. So Yemenis have the right to defend themselves. This is not about greed for Saudi land. But we want the Saudis to feel the pain and suffering that the Yemenis feel from the aggression."
Saudi Arabia claims the Houthis are supported financially and militarily by its regional rival Iran, which Iran denies.
Saudi civil-defence spokesman Colonel Ali Omeer Al Shahrani says Houthi forces have fired rockets across the border.
He says, in one attack, it killed five Saudis and two Yemenis as they were driving past.
"They've hit residential areas, schools and hospitals. The rockets were targeted at civilians, and most of the victims were children."
A Saudi schoolteacher in the city of Najran, Manea Al Ghobari, says children have been victims of Houthi attacks.
"They can't hit our army or our soldiers at the border, so they're hitting civilians in their homes instead. Even today, innocent kids of three, four and nine years are dying because of Houthi shelling on Najran."
A spokesman for the United Nations secretary general, Stephane Dujarric, has called on all sides in Yemen's military conflict to recommit to an immediate cessation of hostilities.
He says then peace talks can resume.
"The Special Envoy insisted that the Yemeni parties should first renew their commitment to an immediate cessation of hostilities and facilitating humanitarian access and engage in discussing a single comprehensive and sequenced agreement for both security arrangements as well as political arrangements."
The United Nations says the 17-month war has left 6,600 people dead.
About 2.5 million Yemenis have also been displaced.
