Android Studio not starting: Fatal error initializing 'com.intellij.util.indexing.FileBasedIndex

This works without the loss any settings or project. It will take you to your previous state at the time of editing an open file.

1. Go to your home directory. ie /home/XXXXXX/.AndroidStudio.X.X
2. Rename the .AndroidStudio.X.X to any thing else i.e. back_up
3. Run your android studio.
4. It will prompt you to import current setting or create a new version
5. choose the import setting and select the back_up directory.
6. Bravo you are good to go.

The Easiest solution that works for the latest Android Studio:

1. Check the latest Plugin [check by date and time] installed in Android Studio C:\Users\{Username}\.AndroidStudio\config\plugins directory. {Mostly Third-party Plugin is the cause}
2. Go Forward Delete that plugin [.jar file]. - no need to delete/rename the whole directory as suggested by others.
3. Close Android Studio properly - through Task Manager or through Restarting your Computer.

Problem solved !!! Cheers.

There was a similar issue posted on Androids open source bug tracker: https://code.google.com/p/android/issues/detail?id=74458

The solution was to restart your computer. Try this and see if it works.