How can I synchronise and/or backup my library between all my devices?

We added support for trakt.tv in the app just for that purpose. If you don't have an account yet, create one, it's free and awesome.
Every update action you take in MOVIST will be automatically pushed to trakt if you are connected. You can also "pull to refresh" every list in the app to fetch modifications done on other devices.

For the synchronisation to work, you will need to have a working internet connection.

Feedback and Knowledge Base