In the Central Kabul several rockets landed near the heavily fortified Green Zone where many embassies and international firms are based, officials said.
There were no immediate reports of any casualties but the explosions occurred in densely populated parts of the Afghan capital.
Kabul police spokesman Ferdaws Faramarz confirmed “multiple rockets” had been fired, while photos on social media appeared to show damage to an external wall at a large medical complex.
The interior ministry said two small “sticky bomb” explosions had been reported earlier Saturday morning, including one that hit a police car, killing one policeman and wounding three others.
No group immediately claimed responsibility for blasts, which come amid an ongoing wave of violence that has wrought carnage across Afghanistan in recent months.
The Taliban have pledged not to attack urban areas under the terms of a US withdrawal deal, but the Kabul administration has blamed the insurgents or their proxies for other recent attacks in Kabul.
Taliban and Afghan government negotiators launched peace talks in Doha in September but progress has been slow.
Officials told AFP on Friday however that a breakthrough was expected to be announced in the coming days, and the US State Department announced late Friday that Secretary of State Mike Pompeo would meet negotiators from the Taliban and the Afghan government in Doha.
Trump has repeatedly vowed to end “forever wars,” including in Afghanistan, America’s longest-ever conflict that began with an invasion to dislodge the Taliban following the September 11, 2001 attacks.
President-elect Joe Biden, in a rare point of agreement, also advocates winding down the Afghanistan war although analysts believe he will not be as wedded to a quick timetable.
Earlier this week, the Pentagon said it would soon pull some 2,000 troops out of Afghanistan, speeding up the timeline established in a February agreement between Washington and the Taliban that envisions a full US withdrawal in mid-2021.
In the past six months, the Taliban carried out 53 suicide attacks and 1,250 explosions that left 1,210 civilians dead and 2,500 wounded, interior ministry spokesman Tariq Arian said this week