Your progress is saved in the Cloud (NOT in Facebook). In order to save your progress and be able to restore it in the future, you need to save your progress in the cloud.
 
For iOS users:
  1. Tap on the Settings icon in the main menu.
  2. Tap on the iCloud switch to turn it green.
 
For Android users:
  1. Tap on the Settings icon in the main menu.
  2. Tap on the controller icon to turn it green.
 

Can I save multiple accounts on one device to play King of Thieves?

Saving multiple accounts by using one iOS or Google account is not possible.

Do not log under different iOS or Google accounts on one device to use different King of Thieves accounts simultaneously. These actions may delete your main progress, and restoration will not be possible. You can only have one KoT account on one device.

It will be extremely helpful if you'll save somewhere your Player ID, so the Support Team can restore your progress manually if anything goes wrong. Your Player ID is a mix of the chosen nickname and numbers, i.e., Thief#12345. You can find it in the Options. You can tap on the arrow icon next to the Player ID to copy it.