The bombs appeared to be aimed at buses carrying security forces and government members, police in Kabul said on Wednesday.
In the first attack, at around 7am local time, a remote-controlled bomb exploded next to an Afghan national army bus in the centre of Kabul, the defence ministry said in a statement.
The blast sent the bus veering out of control and it crashed nearby, wounding 39 people who were taken to a military hospital, the statement said.
Police said the bomb had been hidden in a rubbish bin on the street.
A second bomb, hidden in a cart, exploded 15 minutes later in northern Kabul as a bus carrying employees from the commerce ministry drove past, killing one bystander and injuring several others, police and witnesses said.
A third explosion in an eastern part of the city hit an Afghan army convoy, but there were no apparent casualties.
On Tuesday, two similar explosions in the Afghan capital injured at least seven people.
Violence by Taliban rebels has increased in recent months, with the worst incidents in the south and east of the country where thousands of foreign troops are based.