Can I use my smartphone/tablet with Zwift?

Yes!

Zwift (iOS only)

If you have an iOS device, you can download Zwift from the App Store.

Zwift Companion app

You won't want to ride without the Zwift Companion app! It's currently available on the Google Play Store (Requires Android 4.2 and up) and the iOS App Store (Requires iOS 8.0 or later). See the linked article for details on all it can do.