Cyclone Mahasen has stricken Bangladesh’s southern coast, as people packed into evacuation shelters.
The storm hit Patuakhali district on Thursday with winds of up to 100 km/h (60 mph), and was heading for the ports of Chittagong and Cox’s Bazar.
Some two people have been killed, Bangladeshi officials say.
Around one million people have been ordered to evacuate low-lying areas in Bangladesh and Burma, and take shelter in cyclone centres.
Displaced Rohingya Muslims, living in camps on both sides of the border, may be particularly vulnerable.
But some of those on the Burmese side have resisted calls for them to evacuate camps in Rakhine state.
The UN has warned that 8.2 million people could be at risk from Mahasen in Bangladesh, Burma and north-east India.
At least 956,672 people have been evacuated from Bangladesh’s coastal areas to more than 3,200 cyclone shelters, the government said on Thursday.
The measures in place are a sign of the preparation the authorities have made for the storm.
Warning messages went out to the population across all media before Mahasen hit.
Airports in Chittagong and the resort town of Cox’s Bazar have been shut until the danger subsides. Chittagong’s port, the busiest in Bangladesh, also remains closed.
The Bangladeshi authorities earlier raised the danger level to seven out of 10 for low-lying areas around Chittagong and Cox’s Bazar.
The cyclone covered more than 175km in nine hours before hitting Bangladesh, the country’s Meteorological Department said.
However, the service’s deputy director, Shamsuddun Ahmed, told AFP news agency the cyclone was not expected to cause serious damage as it was “not severe”.
The cyclone “did not gain strength in the last part of its journey as it hit the coast”, he said.
In Bangladesh, there have been reports of waist-deep water submerging low-lying areas and houses being damaged. Dozens of huts collapsed when the cyclone struck Patuakhali district, eye witnesses told local media.
There are also fears of a storm sea surge, and authorities have warned that heavy rainfall could cause landslides in hilly regions.
All schools, colleges and some hotels have been declared cyclone shelters. The centres are crowded and people are still rushing in.
In Burma, meanwhile, tens of thousands of Rohingya Muslims living in camps in low-lying areas of Rakhine state are feared to be at risk.
They were displaced by ethnic violence last year and many are reluctant to move from the camps.
Hla Maung said he lost his mother and two young daughters during the clashes between Muslims and Buddhists.
“I lost everything. I don’t want to go anywhere. I’ll stay here. If I die, I want to die here,” he said.
Rakhine state said it had moved some 36,000 internally displaced people (IDPs) from camps, said Kirsten Mildren, from the UN Office for the Co-ordination of Humanitarian Affairs (OCHA).
But she said the evacuation was “not moving as fast as we’d like – it’s certainly a race against time. We’re finding it very difficult to convince [people] to move to higher ground or safer buildings”.
Burmese planning minister Tin Naing Thein said that in all more than 166,000 people had been relocated, but there was little evidence of a mass evacuation in reports from the affected area.
Correspondents say the Burmese evacuations are seen as a test of the government’s resolve to assist the Rohingya, amid allegations that state forces stood by or even participated in last year’s anti-Muslim violence.
Cyclone Mahasen has already taken a toll. Though the storm did not make landfall in Sri Lanka, the associated heavy rain caused floods and mudslides which killed at least seven people, according to the country’s Disaster Management Centre.
At least 50 Rohingya Muslims drowned on Tuesday when boats evacuating them from the path of the cyclone capsized off western Burma.