I'm looking for software that runs on Linux that can back up my important files. I'm not really talking about system-level backups, since I can always reinstall the operating system. I use Fedora, and therefore reinstall at least every year or so anyway. I'm more interested in backup software that can backup my data files: music, movies, and especially photos, plus other documents and such in my home directory. Ideally, this backup software would intelligently handle external drives that are only connected intermittently (so I can have disconnected backups). It would be nice to have redundancy so that I could have multiple copies of the important data on different areas of the file system. I also think it would be ideal for this software to handle corrupted files intelligently: if a file gets corrupted on one of the copies it would be detected and restored from one of the other copies. I'd also be able to write backups to bluray media so files could be read directly from the media.
I would like to suggest BackInTime. It has the concept of snapshot, thus it will only back up changed data, relative to your reference backup. Not sure about Bluray disks, personally I use external hard drives encrypted with luks.
BorgBackup is a really neat option that offers deduplication of your backups so each subsequent backup only stores what has changed since the last backup, allowing you to store terabytes worth of backups.
This does have the downside where if a particluar file in the backup gets corrupted/lost, its is lost in every backup, but if you run a new backup containing that file, it can restore itself.
Its fairly command-line focused but their documentation is pretty thorough and there's explanations for how to do all sorts of things like start a backup when a drive is connected