More than two million people in India and Bangladesh were evacuated when Cyclone Bulbul hit the Bay of Bengal.
The storm made landfall at midnight local time (1830 GMT) on Saturday near the island of Sagar in West India's Bengal, and is expected to result in waves of up to 2m.
Two people have already been killed by the cyclone, local media reports.
Services at many seaports and airports in the region have also been suspended, including at the busy Kolkata airport.
Bangladesh's two largest ports, Mongla and Chittagong, have been closed and flights to Chittagong Airport have been shut down.
Shah Kamal, Bangladesh's secretary for disaster management, told the US news agency that evacuated residents had been transferred to more than 5,500 cyclone shelters.
Mamata Banerjee, chief minister of the Indian state of West Bengal, tweeted before the hurricane landed urge people to stay calm.
"Please do not panic," she writes. "Please remain calm and cooperate with the administration in its rescue and rescue efforts, be careful, beware and stay safe."
Forecasters expect the storm to move northward and weaken gradually.
It is expected that the wind speed will reach 120 km / h, with gusts of 150 km / h, and create tidal waves in the sea and the rivers when it strikes the coastal regions, announced the Department of Meteorology of Bangladesh.
The Sundarbans, the largest mangrove forest in the world and home to Bengal tigers, are endangered.
But the closure of the transport centers has left thousands of other people stranded on islands off the coast, including the island of St. Martin in Bangladesh.
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The Indian authorities have stated that ships and military planes have been put on hold to respond to emergencies.
The lower coast of Bangladesh is often hit by deadly cyclones, but the country has managed to reduce the number of casualties in recent years.
Early warning systems for cyclones have improved, giving the authorities more time to evacuate people. Other anti-cyclone shelters have also been built to protect local residents.