They're particularly cross at people who run and use sites set up to swap music with other people - called 'peer-to-peer sites' in the US.
'Out of their minds'
Anyone who is caught making this music available, or downloading it, will be tracked down and taken to court.
People caught will have to face £90,000 law suits against them, the Recording Industry Association of America has pledged.
They say the music industry has lost loads of money, and record sales have plummeted, purely because of illegal downloading.
But the president of one American music-swapping site said: "They're out of their minds."
He said the lawsuits would just make music fans upset with the big record companies who are behind it.