The so-called Islamic State group (IS) has said it carried out the attacks.
The group has established a presence in Yemen since a civil war broke out there and has carried out numerous attacks.
It is opposed to the government and Shia Houthi rebels who have seized much of the country, including the capital Sanaa.
The attacks come on the eve of the first anniversary of the Saudi Arabia-led coalition against the rebels.
Both civilians and military personnel were killed by the blasts, officials said. Dozens of others were wounded.
Two of the bombings hit checkpoints next to a base used by the coalition, they said. Gunmen then tried to attack the place.
The third explosion happened when an ambulance laden with explosives was detonated next to a military checkpoint.
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Aden was recaptured by government troops backed by the Saudi-led coalition last year.
The campaign was launched in March 2015 in an attempt to oust the rebels from Sanaa and restore Yemen's government.
Riyadh says the Houthis, who are allied with forces linked to the former president, are supported militarily by its regional rival Iran, something it denies.
Since the offensive started, more than 6,200 people have been killed, half of them civilians, according to the UN. The war has left Yemen fragmented and facing a humanitarian crisis.
Meanwhile, IS and Al-Qaeda in the Arabian Peninsula (AQAP) have exploited the conflict to expand the territory under their control.

But the BBC drama - which celebrated its 30th anniversary last month - was beaten to the TV soap of the year award by its ITV rival Coronation Street.
Ant and Dec were named TV personalities of the year for the third year running at the industry bash in central London.
Susanna Reid, meanwhile, won best news presenter/reporter for the second year in succession.
The event saw a number of last year's award recipients recognised again, among them Strictly Come Dancing (best reality programme), Channel 4's Gogglebox (best entertainment programme) and BBC Breakfast's Carol Kirkwood (best weather presenter).
Other winners included BBC One's popular teatime quiz Pointless (best daytime programme), Silent Witness (best crime programme) and Downton Abbey, which was named best drama over its ITV stablemates Cilla and Doc Martin.
Capital Radio's breakfast show was named best digital radio programme, while Radio 2's Chris Evans won the digital radio personality prize.
The event also saw the BBC's football highlights show Match of the Day honoured with a special Tric award.
Dyer's EastEnders character, Mick Carter, made his first appearance on Albert Square on 25 December 2013.
Speaking to chat show host Graham Norton last month, the 37-year-old Human Traffic star said he was "happy to sit back and be part of" a show he likened to "a retirement home".
First presented in 1989, the Tric Awards are voted for by members of the TV and radio industries.

The England and Wales Cricket Board made the move in an attempt to improve the standard of pitches and Rhodes says it has already made a difference.
"I think it has," he told BBC Hereford and Worcester.
"If you look back, there have been some very poor wickets - maybe this was the only way they could do it."
The 51-year-old former England wicketkeeper added: "I think, to a certain extent, it's worked.
"There have been some criticisms about wickets being too flat but if you're prepared to bat for a long period of time and concentrate, you'll get runs."
Worcestershire's two Championship matches so far  - which have both been draws - have produced eight centuries, with five coming in the match against Gloucestershire and three more in their last game with Essex.
England all-rounder Moeen Ali, Joe Clarke, Brett D'Oliveira and Tom Kohler-Cadmore have all reached three figures for the New Road side and Rhodes says better pitches will help sort out the best players.
"Too many good players were playing forward defensive shots and getting out to average bowlers - that's what was happing before with the wickets," he added.
"Now those bowlers are struggling a little bit - which I think is a good thing."
At least 22 people have been killed by three suicide bombings in checkpoints in the southern Yemeni city of Aden, officials say.
[NEXT_CONCEPT]
EastEnders' Danny Dyer has been named soap personality of 2015 at the TV and Radio Industries Club (Tric) awards.
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Worcestershire director of cricket Steve Rhodes says the decision to scrap the toss in the County Championship this season looks like it is working.