The army has been deployed to help with search and rescue efforts in the affected hill region of Chittagong, said the region's chief administrator, Sirajul Haq Khan, who warned that the toll could rise.
According to Khan, at least 26 people died in a series of landslides and flash floods in and around Chittagong port and the district of Cox's Bazar.
In neighbouring Bandarban district, 30 bodies have been recovered from multiple landslide sites, local administrator Tariqul Islam said.
"Rescue efforts had been hampered as communications have been largely snapped because of flash floods and heavy rain," Islam told AFP.
"Ten of the victims were children and scores of others were injured," he said.
Bandarban police chief Saiful Ahmed said most of the victims were asleep when the huge chunks of mud buried them alive.
"One family has lost 12 members," Ahmed said.
Chittagong port received 40 centimetres (16 inches) of rain in a single 12-hour period on Tuesday.
Flights in and out of Chittagong's Shah Amanat International Airport have been suspended since Tuesday afternoon.
According to the state Disaster Management Information Centre, around 50,000 people were affected by the flash floods, and many of them forced to take shelter on higher ground.
Train links between Chittagong and the rest of the country were also severed after a railway bridge collapsed due to a rain-triggered flash flood.
In recent years, monsoon rains have caused deadly landslides in Chittagong, home to five million people, killing hundreds and prompting the government to tighten rules on where development can take place.
In June 2007, landslides in Chittagong killed at least 130 people. Another 53 were killed by to flash floods and landslides in Cox's Bazar district in 2010.
What the stars mean:
★ Poor ★ ★ Promising ★★★ Good ★★★★ Very good ★★★★★ Exceptional