At least 11 bombers targeted the city of Zaranj, police said, but not all had been able to blow themselves up.
Shortly afterwards, police in the northern province of Kunduz said 12 people were killed by another bomb.
The bombers had reportedly attacked crowded markets.
The district governor of Dashte Arche district in Kunduz said the bomb had been placed on a motorbike and had gone off shortly before the end of the day’s fast for the Muslim holy month of Ramadan.
Many of the victims were thought to be civilians, including food-sellers, he said.
In the south-western province of Nimroz, deputy police chief Mujibullah Latifi told AFP that some of the attackers had been killed by police.
“There have been heavy casualties; the majority of them are civilians,” he said.
Reports say two of the bombers’ explosives were detonated when police fired on them.
Afghan intelligence officials have said a number of potential suicide bombers infiltrated Zaranj.
Some were arrested on Monday, with further arrests early on Tuesday.
One official said he had been searching for further suspected insurgents in a crowded marketplace when the attacks began.
An eyewitness, Mohammad Zalmay, said: “I was buying sweets with my sons and daughters when I heard a bang. I fell to the ground. When I woke up, I saw blood all over.”
Zaranj, near the Iranian border, is a relatively affluent and peaceful city.
The police will be extremely concerned that there will be further attacks, he says.
International troops are gradually handing over responsibilities to Afghan security forces, as NATO prepares to pull out of the country by the end of 2014.
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