One explosion occurred at a police checkpoint outside a court building in the port city of Chittagong.
The blast occurred just before the court opened at 9am (0300 GMT).
The second explosion was in Gazipur, 30km north of the capital Dhaka, when a powerful bomb went off in a court complex, local hospital doctor Shafiqul Islam said.
Eight people, including two bombers, were killed and 23 people – including five police officers and a would-be bomber – were in a serious condition in hospital, police said.
Scores more at the blast sites suffered minor injuries.
The explosions in both cities occurred at about the same time.
Habibur Rahman, a doctor at Chittagong Medical College hospital, said 16 people, some of them in a critical condition, were being treated at the hospital.
Bangladesh has been hit by a spate of bombings blamed on banned Islamic group Jamatul Mujahidin Bangladesh. The group seeks to establish Islamic rule in the Muslim-majority country, which is governed by secular laws.
Two weeks ago two judges were killed in the southern town of Jhalakathi by an attacker also allegedly linked to the banned group.
Another four people were killed and dozens injured in multiple bomb attacks in August and October.
The attacks targeted government buildings and courts.
Police accused the Jamayetul Mujahideen of staging the attacks targeting the nation’s legal system.
“We are stepping up security around the country,” said national police chief Abdul Kaiyum after the twin attacks.