Hurricane Gustav hit Cuba's Isle of Youth and moved towards the mainland on Saturday as forecasters said it could grow into a Category 5 storm.
Gustav was a Category 4 storm with winds of 230kph as it crossed the Isle of Youth, which has 86,000 residents, just off southwest Cuba.
Heavy rains and strong winds are reported over Cuba's western province of Pinar del Rio, the main tobacco-growing region.
In its latest advisory, the US National Hurricane Center said the storm was centered 135km southwest of Havana and moving northwest at 24kph.
Power was out across most of Havana. There were no immediate reports of injuries or deaths.
Recent harvests of tobacco crops have been moved to safe places and Cuban officials said at least 200,000 people had been evacuated from the path of the storm.
In Havana, residents have been warned to take precautions.
Hurricane deaths are rare in Cuba, but Gustav killed at least 86 people in the Dominican Republic, Haiti and Jamaica. No deaths were reported in the Cayman Islands overnight.
Gustav could reach the coast of Louisiana early on Tuesday.
People were lined up outside a bus and train terminal early on Saturday to get out of New Orleans before it arrives.
Cars are reported to be bumper to bumper on highways leading out of the city and six low-lying parishes - the Louisiana equivalent of counties - have issued mandatory evacuation orders effective later on Saturday.
New Orleans marked the third anniversary of Hurricane Katrina on Friday. However, the city has not yet issued a mandatory evacuation order this time.
Mayor Ray Nagin said a mandatory city evacuation could start early on Sunday if Gustav holds to its current course.
Katrina hit the coast near New Orleans as a Category 3 hurricane on 29 August, 2005, with wind speeds up to 209 kph.
Its storm surge broke through levees and flooded 80% of the city. About 1,500 people were killed on the US Gulf Coast. Damage amounted to $US80 billion.