No one immediately claimed responsibility for the attack but the Islamic State group affiliate in Afghanistan has claimed several brutal attacks on Shiites in recent months in Kabul.
It has scaled up its attacks in Kabul, including on security installations and the country's Shiite minority.
A terrorist bombing on a Shiite cultural center in Kabul on Thursday prompted a renewed pledge from the United States to continue backing Afghanistan's government until the nation is safe and at peace, US Ambassador John Bass said in a statement.
AFP reporters near the scene of the blast heard a third explosion, suggesting the attack may still be under way.
Later, Wahidullah Majrooh, a spokesman for the Health Ministry, said that 35 bodies and 20 wounded people had been taken to a hospital so far.
President Ashraf Ghani's spokesperson condemned the "unpardonable" crime, and his office has ordered a full investigation into the attack.
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"Terrorists always hide their defeat against our security forces by attacking mosques, religious places, and educational institutions, and killing our people, including women and children", Ghani wrote on Facebook.
However, no-one has taken responsibility for the latest attack that killed so many people.
The IS group statement said the attack was carried out by one bomber wearing a vest and followed three other bomb blasts in the same area.
He called them attacks as against Islam and "all human values". Before Thursday's attack, two journalists and five media assistants were killed doing their jobs this year in the country, according to a report this month by Reporters Without Borders, an organization that works to protect freedom of press.
Interior Ministry spokesman Najib Danish says an unknown number of suicide attackers set off an explosion outside the center before storming it.
Dawlat Abad District Gov. Mohammad Karim said Thursday the powerful mine killed six shepherd children Wednesday ages 10, 9 and 8.