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This function isn't entirely new - you've been able to simply view an area of a map offline for a while - but now Google gives you turn-by-turn driving directions, the ability to search on an offline map, and local business information.
Here's how to use the new feature:
First and foremost, download the Maps app. Then, search for a location, like "San Francisco" or "Tokyo." You'll see a "Download" button appear alongside information about the place:

Next, Google lets you specify exactly what parts of the map you want to save. You can only save 2500MB per map, but you can save multiple maps of the same location, if everything you want doesn't fit:

You'll be prompted to name the map, and then Google will download the segment. Here's what that process will look like:

You can then find that segment - as well as any other Maps you saved - by swiping right on the app or pressing the three horizontal lines in the Maps search bar and tapping "Your Places":

Google will save each Map for 30 days, and the app will automatically move into offline mode if you're in an area with spotty service or no connectivity. You can still search for specific places on the offline map, and get information such as hours of operation and customer ratings:
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... As well as turn-by-turn directions:
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And voila! You're all set to explore the world without cell phone service.
Although these features are rolling out now for Android, iPhone users will have to wait a bit longer - Google says the changes are coming soon to iOS.