How do I sync my aircraft profiles and flight plans across devices?

This is very simple. Go to Settings > User Settings > Sync Flight Plans and/or Sync Aircraft.

Make sure you do this on each device to enable syncing.

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Aircraft syncing general advice:

When preparing to sync to a new device, it's important to make sure that the snapshot of your aircraft up in your section of AvPlan Cloud is the most up-to-date one.  If you don't, there is a possibility that you can overwrite your current aircraft stored on your device.  To ensure you avoid this situation, take the following steps:

Open the device that you are sure has the correct aircraft set up on it.

Tap Settings > User Settings and ensure that Sync Aircraft is ticked. Then,

Tap Settings > Aircraft Type Database.  

Tap on an aircraft you have set up and make one simple change (One that won't have a large effect on the flight planning you've already set up - amending the price of Fuel up or down by one cent is enough for the purposes of syncing).  

Once you've made the alteration, tap the back button (  < symbol in the top-left of the window) two or three times - until you're prompted to save or discard your changes.  Tap Save.

AvPlan EFB will now push up an up-to-date snapshot of your aircraft to the cloud.

Now, you can pick up your other device(s) and do the following (assuming you're logged in):

Tap Settings > User Settings and ensure that Sync Aircraft is ticked.

Tap Settings > Aircraft Type Database.

Tap the circular arrow icon in the bottom-left corner of the screen.  The app should say 'Restoring' for a few seconds.

The aircraft from your first device should now be present on your new device.

NOTE:  If you tap the circular arrow icon without a snapshot of your aircraft saved in the cloud, the current saved aircraft on that device may be overwritten with no aircraft.