The federal court case in the US is being brought by Epic Games, maker of the popular video game Fortnite.
Why is Epic Games taking action?
Epic claims that Apple has transformed its digital storefront into an illegal monopoly that squeezes mobile apps for a significant slice of their earnings.
Apple, which denies Epic's claims, started building the App Store 13 years ago as part of a strategy masterminded by co-founder Steve Jobs.
Experts say the trial is "enormous" and has the potential to change the way Apple works.
It could give more power to app developers - or cement Apple's control on the App Store.
How valuable is the App Store to Apple?
Apple takes a commission of 15% to 30% on purchases made within apps, including everything from digital items in games to subscriptions.
The store feeds more than 1.6 billion iPhones, iPads and other devices and has helped turn Apple into one of the world's most profitable companies.
It boasts a market value that now tops 2.2 trillion dollars (£1.6trn), while privately-held Epic is far smaller in comparison, with an estimated value of 30 billion dollars (£21.7bn).
What is the background to this trial?
Epic says it handed out hundreds of millions of dollars to Apple before the technological giant expelled Fortnite from its App Store last August, after Epic added a payment system that bypassed Apple.
Epic then sued Apple, prompting a courtroom drama that could shed new light on Apple's management of its App Store.
Are there any other motives for Epic's action?
Epic, based in North Carolina, is looking to offer an alternative app store on the iPhone.
It also wants to break free of Apple's commissions.
What can we expect from the trial?
Both Apple chief executive Tim Cook and Epic chief executive Tim Sweeney will give evidence in a federal courtroom in Oakland, California.
Neither side wanted a jury trial, leaving the decision to US District Judge Yvonne Gonzalez Rogers, who already seems to know her ruling will probably be appealed against, given the stakes in the case.
Epic contends the iPhone has become so ingrained in society that the device and its ecosystem have turned into a monopoly Apple can exploit to unfairly enrich itself and thwart competition.
Apple claims it faces significant competition from various alternatives to video games on iPhones.
Epic has filed a separate case against Google, accusing it of illegally gouging apps through its own app store for Android devices.
Apple will also try to depict Epic as a desperate company hungry for sources of revenue beyond the aging Fortnite.
The trial is expected to last for most of this month, with a decision expected in the following weeks.