Yup! Except don't delete the old version if you're on Android - just overwrite the old apk with the new one.
PC/Linux/Mac: Download update to a new folder and play!
Android: Download update and install as normal
The game will automatically transfer the save data over. If you're worried, then don't delete the old PC/Linux/Mac version until you've turned on the new one and made sure the saves are there. It won't do any harm to have multiple versions of the game on your hard drive :)