Gen. Mohammed Musa Rasuli said 29 people died and 57 were wounded in Zaranj, the capital of Nimruz province in southwestern Afghanistan, The New York Times reported.
District Gov. Sheik Saadi told the newspaper 10 people were killed when a remotely detonated bomb on a motorcycle went off in a market in Dashte-e-Archi, a district in Kunduz province in northern Afghanistan.
Abdul Rasul Rasekh, a spokesman for the governor of Badakhshan province in northeastern Afghanistan, said a district governor and three police officers died in a Taliban ambush as they drove through a remote area.
Iran's Press TV reported the death toll had reached at least 46.
No group had claimed responsibility for the attacks, the Times and Press TV said.
The Times said the attacks, which came as people prepared for the holiday that ends the Muslim holy month of Ramadan, produced the war-torn country's largest single-day civilian death toll this year.
Afghan President Hamid Karzai, who was in Saudi Arabia, condemned the attacks.
"The terrorists are so desperate and abject that they kill our innocent Muslims in the holy month of Ramadan," Karzai said.
The multiple explosions in Zaranj occurred despite authorities killing two potential suicide bombers Monday night and capturing three other suspects earlier Tuesday.
Rasuli said the first two bombs exploded near the governor's office and the third targeted a police car. Another then went off as people gathered in front of a hospital across the street from a crowded market.
Rasuli said three other suspected bombers had been shot elsewhere in the city.