Texas will be hit by "catastrophic and life-threatening flooding", the US National Hurricane Center has warned, as Tropical Storm Harvey moves inland.
Harvey battered the coast as a category four hurricane - the biggest to strike the US mainland in 13 years. It has now been downgraded.
Some residents are feared to be trapped in collapsed buildings and there have been widespread power outages.
Torrential rains are expected for days. One person has died in a house fire.
The victim, in the city of Rockport, was found after the storm moved inland, officials said.
Up to 40in (101cm) of rain could fall in some areas of the middle and upper Texas coast, the hurricane center (NHC) warned.
Several places are already reported to have received well over 10in.
Coastal areas will also be flooded by storm surges during high tide, the NHC said.
Earlier, Harvey made a double landfall: north-east of the city of Corpus Christi initially late on Friday local time, then just north of Rockport a few hours later.
When the storm struck close to Corpus Christi it had winds of up to 130mph (215km/h), but by 18:00 GMT its sustained winds had dropped to 70mph - therefore becoming a tropical storm.
It was moving at an extremely low speed - 2mph.
Texas Governor Greg Abbott told a news conference that there had been 20in of rain in Corpus Christi and 16in in Houston - but another 20-30in was expected on top of that.
Utility companies say nearly 300,000 customers are without electricity.
The BBC's James Cook was in Corpus Christi when the hurricane hit. Although the wind was screeching all night, he says, in the morning there was no sign of any serious structural damage there.
President Donald Trump has freed up federal aid for the worst-affected areas and Governor Abbott praised the administration for "stepping up".
In a teleconference with cabinet members Mr Trump "emphasised his expectations that all departments and agencies stay fully engaged and positioned to support his number one priority of saving lives", the White House said.
At least 18 people have been rescued from vessels in distress by Coast Guard helicopters, the Associated Press reports. ABC News has tweeted a video of four people being rescued off the coast of Port Aransas.
Rockport, normally home to about 10,000 people, appears to have been the hardest hit town.
The roof of a housing centre for elderly people there collapsed, trapping people, 10 of whom were later taken to a county jail for treatment, according to local TV station KIII.
Emergency services said a school, a hotel and other buildings had also suffered structural damage.
Before the storm arrived, Mayor Patrick Rios had a stark warning for anyone who refused to leave town: "We're suggesting if people are going to stay here, mark their arm with a Sharpie pen with their name and Social Security number."
When the storm struck near Rockport, wind speeds were 125mph.
Authorities have not been able to inspect the full extent of the damage.
Travis Pettis, a reporter with the Caller Times newspaper in Corpus Christi, told the BBC that strong winds when the storm arrived made the rain feel like "needles".
"It was pretty crazy out there. The winds, I don't know what [speed] they topped at but you could barely even walk," he said.
Trees came down in the city and power supplies were cut, but thousands of residents had heeded warnings and boarded up their homes before fleeing.
Port Aransas, a city on Mustang Island, near Corpus Christi, is also reported to have extensive damage. Local media reports say a search and rescue operation is under way at a trailer park.
Mr Abbott said it was obvious Texas was confronting a "very major disaster" and more than 1,800 military personnel will help with the clean-up.
Harvey is the first major storm of the Atlantic hurricane season. It is forecast to meander across south-east Texas until Wednesday.
Oil-rich Houston, the fourth biggest city in the US, could face up to 20in of rain over the coming days.
Residents there should "not be complacent" and should stay at home and "ride out the storm", Francisco Sanchez, a spokesman for Harris County's emergency management office, told the BBC.
"The most severe weather is yet to come" in the Houston area, he said.
About 4,500 inmates from prisons south of Houston are being evacuated to other prisons in east Texas because the Brazos River nearby is rising significantly, officials say.
Mr Trump is likely to visit Texas early next week, the White House said.
The storm is the strongest to hit the US since Charley in August 2004 and the most powerful to hit Texas since Hurricane Carla in 1961, which killed 34 people, according to the Houston Chronicle.
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