Hurricane Sandy has killed at least 21 people in the Caribbean, but weakened as it reached the central Bahamas last night.
State media in Cuba said Sandy toppled houses, ripped off roofs and killed 11 people in the eastern provinces of Santiago and Guantanamo.
Nine deaths were reported in Haiti and one in Jamaica.
By late last night, Sandy had slowed to a category one hurricane, but forecasters warned it will likely blend with a winter storm to cause a super storm in the eastern US next week.
They said its effects will be felt along the entire Atlantic Coast and inland to Ohio.
Some further weakening in Sandy has been forecast for the next 48 hours, but it is expected to remain a hurricane.
Late last night the hurricane's centre was 300km east-southeast of Freeport on Grand Bahama Island as it spun between Cat Island and Eleuthera in the central Bahamas.
The storm's maximum sustained winds had fallen to 150kph.
Caroline Turnquest, head of the Red Cross in the Bahamas archipelago off Florida's east coast, said 20 shelters were opened on the main island of New Providence.
"Generally people are realising it is serious," she said.
Power was out on Acklins Island and most roads there were flooded, government administrator Berkeley Williams said. He said his biggest concern was that a boat filled with basic supplies for the island had to cancel its trip until next week.
"Supplies were low before, so you can imagine what we are going through now," Mr Williams said.
Cuban authorities said the 11 dead included a four-month-old boy who was crushed when his home collapsed and an 84-year-old man in Santiago province.
It was Cuba's deadliest storm since July 2005, when Hurricane Dennis slammed into the island as a category five storm, killing 16 people.
Santiago, Cuba's second largest city, near the eastern tip of the island, was spared the worst of Sandy, which also slammed the provinces of Granma, Holguin and Las Tunas.
Cuban President Raul Castro ordered authorities to evaluate damage throughout eastern Cuba.
In Haiti, Joseph Edgard Celestin, a spokesman for the civil protection office, said the country's death toll stood at nine, including three people who died while trying to cross storm-swollen rivers in south-western Haiti.
He did not provide specifics of how other people died.
Officials reported flooding across Haiti, where many of the 370,000 people still displaced by the devastating 2010 earthquake scrambled for shelter.
More than 1,000 people were evacuated from 11 quake settlements, according to the International Organization for Migration.
Sandy was blamed for the death of an elderly man in Jamaica who was killed when a boulder crashed into his clapboard house, police said.