Men on a motorbike threw three grenades in the symbolic heart of the city, while two other blasts were reported in a northern suburb, according to witnesses and a journalist group, SOS Medias Burundi.
A young child was killed, with SOS Medias Burundi issuing a photograph of the victim, a small corpse covered by a blanket lying on the street.
Public Security Minister Alain Guillaume Bunyoni confirmed a child was killed and 30 people wounded.
Security forces, rebels and opposition all blame each other for the killings.
Presidential press chief Willy Nyamitwe confirmed the attacks, saying that "as usual... Terrorists kill civilians with grenades."
Burundi plunged into crisis in April when President Pierre Nkurunziza ran for and won a controversial third term, sparking street protests, a failed coup, regular killings and a nascent rebellion.
Hundreds have been killed in the violence and at least 230,000 have fled the country.
"We do not know what is going on. But it is clear they want to terrorise us. I heard three grenade explosions and here, near the central roundabout, there were several wounded," said one witness, who asked not to be named.
The rebels formed the force "to protect the population" and uphold the Arusha Agreement that paved the way to the end of the country's 1993-2006 civil war -- an accord they say Nkurunziza violated by his third term in power.
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