The Syrian Observatory for Human Rights, which said regime warplanes carried out the air strikes, earlier gave a toll of 19 dead but said this was likely to rise.
An AFP correspondent in the opposition-held eastern part of the city said several districts were targeted including Bustan al-Qasr, Al-Mashad and Salhin.
Late today, two barrel bombs also hit the southern district of Bab al-Nayrab, killing an ambulance driver.
"It was a bloody, black day in Aleppo, with aircraft not leaving the sky," Bibres Mashaal, the head of civil defence in Bab al-Nayrab, said.
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"Today the civil defence was fully mobilised and there were also fires," Mashaal said.
Civil defence workers also reported three people killed by artillery fire in Bustan al-Qasr, one of Aleppo's most heavily populated neighbourhoods.
An air strike there earlier hit a five-storey apartment building, shearing off part of an entire floor.
"We were sleeping at 10:00 am when the strike hit the fourth floor of the building," said resident Ahmad Radi.
"We ran down and found the bodies all over the ground."
"It's become normal here for people to die every day. No one even mourns anymore," one Bustan al-Qasr resident said.
"The next day, everyone opens their shops and things carry on as if nothing happened. But everyone living here has lost someone."
"A surprising number of wounded showed up at the field hospital, around 20 people," one medic in an opposition-held neighbourhood told AFP.
"It's more than we can handle," he said, adding that field hospitals in other neighbourhoods were also struggling to cope.
Nearly all warring parties in Syria -- the regime, rebels, jihadists and Kurds -- have carved out zones of control in war-torn Aleppo province.
A ceasefire took effect in Syria at the end of February, but the country has been rocked by fighting in recent weeks, particularly around Aleppo.
Syria's conflict erupted in March 2011 with anti-government protests, but has since spiralled into a multi-front war that has left more than 270,000 people dead.