Cyclone Mora has stuck Bangladesh, lashing the country's south-eastern coast with rain and winds.
The storm made landfall on Tuesday morning (local time) between the fishing port of Cox's Bazar and the city of Chittagong, bringing winds of up to 117km/h.
Authorities have been moving more than one million people to shelters.
India and Myanmar are also expected to be affected. It comes after heavy rains in Sri Lanka caused landslides that killed at least 180 people.
Bangladesh's meteorological department has previously warned that low-lying areas of Cox's Bazar, Chittagong and many other coastal districts were "likely to be inundated" by a storm surge of 1.2m-1.5m above normal levels.
Hundreds of thousands of people had been evacuated to shelters, schools and government offices.
Fishing boats and trawlers have been advised to remain in shelters as well.
Tracking website Tropical Storm Risk has forecast that Cyclone Mora would move northwards past Chittagong over the next 12 hours, weakening as it got further inland, and would be downgraded from a Category-One cyclone to a tropical storm.
No casualties have been reported so far, but the cyclone caused significant damage to refugee camps for Rohingya Muslims, community leader Abdus Salam said.
He said 20,000 houses in refugee camps were damaged.
"In some places, almost every shanty home made of tin, bamboo and plastic has been flattened. Some people were injured, but no-one is dead," Mr Salam said.
Large camps have been set up in Cox's Bazar for hundreds of thousands of ethnic Rohingya who have fled violence from neighbouring Myanmar.
The Bay of Bengal is prone to storms, and Bangladesh is often hit by severe weather during the monsoon season from the middle to the end of the year.
Last year Cyclone Roanu hit coastal Bangladesh, leaving at least 24 people dead.