I don't think many people use Handoff to transfer data between apps, such as in my Mail example, across Apple devices. Partly, that's because OS X makes it easy to detect a potential Handoff, but the notification feature is all but invisible on iOS devices, so it's harder to feed the habit.
What Microsoft's Project Rome does
Microsoft's Project Rome could be used to develop Handoff-like interactions. However, from Microsoft's demos, it appears that the operational model is more like remote control, where the first device initiates the interaction with the second device, contrary to Handoff's approach in the second device initiating the transfer that's offered quietly in the background as long as there is a connection. Also, Project Rome is more generalized than Handoff, offering several types of interactions.
First, it can detect devices both locally over Bluetooth or Wi-Fi and elsewhere via the Internet -- not only locally as in the case of Handoff. Thus, it's useful for some remote desktop-style uses, such as pulling info from your office computer when at a conference or sending a presentation to a computer in another location.
Second, the APIs are aimed at a range what Microsoft calls "app experiences," using a combination of app-launching protocols and communications payloads.
There are APIs for launching an app on another device, along with a Uniform Resource Identifier (URI), to know what content to pull. The notion here is if a user has, say, the Facebook app installed, clicking a link to a Facebook post should open the post in the Facebook app, not at the Facebook website in a browser. The URI is essentially used like an anchor tag would be on a website to figure out what item to display. Via settings, users can control whether such links open the app or the website on a per-app basis.
There are APIs to extend an app interaction to other instances of the app on other devices. For example, you might run an app on your phone, then have it launch on your desktop as well, so you have both apps active and interacting. An example is watching a movie on one device and interacting with a related trivia game on the other based on what part of the movie you are currently watching. This is different than Apple's AirPlay or Microsoft's WiDi screen mirroring, which simply sends the screen to another device -- in Project Rome, the app runs on both devices and the two instances can communicate with each other.
There are APIs to remote-control an app across devices. Apple's Keynote and Microsoft's PowerPoint can already do that, letting you manage a presentation on your laptop or tablet from your phone or (in Apple's case) watch. With Project Rome, Microsoft provides a widely available API for the task, not rely on custom code developed for each app as in the case of Keynote and PowerPoint.
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