(Reuters) - In the span of two days this week, Alphabet's Google was hit with a court order to open up its app store Play to rivals and learned that the U.S. Justice Department may soon ask a judge to split up the company to limit its hold over online searches.

Here is a look at the U.S. antitrust cases that have Google playing defense and that could help shape the company's future.

ANDROID APPS

U.S. District Judge James Donato in San Francisco on Monday ordered Google to overhaul parts of its Android app business, spelling big changes for consumers, developers and mobile device makers. For three years beginning in November, Google under the order must allow Android users to download rival app platforms and use competing in-app payment methods, and it cannot pay device makers to preinstall its app store Play.

The order came in a lawsuit that "Fortnite" video game maker Epic Games filed in 2020 accusing Google of monopolizing Android app distribution and payments. Google said it will appeal an underlying jury verdict in the case and challenge the court-ordered changes.

Google in a related Play case said it will pay $700 million to resolve claims by consumers and U.S. states that it charged inflated prices for apps. That proposed settlement is pending before Donato, who has questioned whether the amount is sufficient.

Separately, Epic last month brought a new lawsuit accusing Google of unlawfully conspiring with device maker Samsung to protect Play from competition. Samsung and Google denied the claims.

ONLINE SEARCH

Google's dominance over online search is the centerpiece of a lawsuit filed by the Justice Department in federal court in Washington in 2020. In that case, U.S. District Judge Amit Mehta ruled in August that Google had established an illegal monopoly by paying billions of dollars to rivals to make it the world's default search engine.

The Justice Department told the judge on Tuesday that it may seek an order breaking up parts of Google to remedy the company's antitrust violations. The government's specific proposals - such as potentially forcing Google to divest its Chrome browser and Android operating system - are due in November.

Google is due to propose its own remedies in late December, and the judge is scheduled to hear arguments in April 2025.

DIGITAL ADVERTISING

Beyond search and apps, Google faces a trio of lawsuits challenging its dominance in the market for online display advertising. U.S. District Judge Leonie Brinkema in Virginia held a non-jury trial last month in a case filed last year by the Justice Department and a group of states. Google is accused of unlawfully dominating all sides of the ad market, forcing customers to use its products and using its size to crush rivals. Closing arguments are scheduled for late November. 

Google is also fighting two related cases in federal courts in Texas and New York.

Texas is leading a group of states suing Google over digital ads in a case scheduled for a March 2025 trial. Publishers and advertisers are pursuing related claims challenging Google's advertising technology practices, alleging they have been overcharged and lost revenue.

(Compiled by Mike Scarcella; Editing by David Bario and Will Dunham)