The biggest storm to hit the US mainland in 13 years is moving slowly, with 16.43in of rain reported in one area.
Catastrophic flooding is expected, though Hurricane Harvey’s winds were downgraded from category 4 to category 1.
Some residents are feared trapped in collapsed buildings and there have been widespread power cuts.
More than 200,000 customers are without electricity, utility companies say.
President Trump has approved a federal aid for the worst-affected areas and praised emergency services in a tweet: “You are doing a great job – the world is watching! Be safe.”
Texas Governor Greg Abbott warned of record flooding in multiple regions after Harvey made a double landfall: north-east of the city of Corpus Christi initially, then just north of Rockport a few hours later.
Rockport, normally home to about 10,000 people, appears to have been the hardest hit. Emergency services there say a number of buildings have suffered structural damage.
Images posted on social media show collapsed trees and houses.
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