In August 2012, a jury found Samsung guilty of infringing six Apple patents and awarded one of the largest payouts of its kind on record.
The verdict and $1bn (£626m) awarded in damages were seen as a massive victory for Apple.
This is just one of many patent cases the two companies are fighting in courtrooms across the world.
In March 2013, a judge re-examined the $1bn awarded to Apple and reduced the amount saying the damages would need to be assessed at a new trial.
Judge Lucy Koh said the original jury in the court in California had incorrectly calculated part of the damages.
The judge said that $550m of the award had been worked out in the proper manner but she ordered that the remaining $450m be reassessed.
That $450m could be increased or lowered meaning that Apple could conceivably end up with more than the original $1bn awarded at the first trial.
Apple had originally sought $2.5bn in damages from Samsung.
It argued that the South Korean company had copied its designs for the bodies of the original iPhone and iPad as well as user-interface elements such as the bounce-back response when a person scrolls beyond the end of list and tap-to-zoom.
Samsung argued it was already working on rounded rectangular handsets dominated by a screen and a single button months before the iPhone was revealed. It sought $519m in damages from Apple.
Stealing isn’t right
At the time of the original ruling Samsung said the decision was bad news for consumers and would “lead to fewer choices, less innovation, and potentially higher prices”.
Apple said it applauded the court “for finding Samsung’s behaviour wilful and for sending a loud clear message that stealing isn’t right”.
Even after this latest trial, which will involve a new jury, both companies could appeal.
“I think they will appeal unless they settle,” intellectual property consultant Florian Mueller told the BBC.
“But at this point neither party has enough leverage to force someone into settling if they don’t want to. And we’re not there yet,” he said.
Apple has asked for a sales ban to be imposed on the Samsung products that had been found to infringe the patents. But the judge ruled that Apple could adequately be compensated financially.
“Between these types of big players a $1bn damages award is not as bad as a sales ban,” said Mr Mueller.
Apple appealed against that ruling and the result of the appeal is due before the end of the year.
Samsung and Apple are currently locked in a battle in the courts of more than 10 countries across Europe