Rsync is a fast and extraordinarily versatile file copying tool. It can copy locally, to/from another host over any remote shell, or to/from a remote rsync daemon. It offers a large number of options that control every aspect of its behavior and permit very flexible specification of the set of files to be copied. It's a build in Linux tool. Real hardcore =)
rsnapshot is a filesystem snapshot utility for making backups of local and remote systems. Using rsync and hard links, it is possible to keep multiple, full backups instantly available. The disk space required is just a little more than the space of one full backup, plus incrementals. Depending on your configuration, it is quite possible to set up in just a few minutes. Files can be restored by the users who own them, without the root user getting involved. Stackoverflow users recommended it to me couple of years ago and I thinks that is a really good solutions, which unites the best from rsync and Linux filesystem.
Snapper is a tool for Linux filesystem snapshot management. Apart from the obvious creation and deletion of snapshots, it can compare snapshots and revert differences between snapshots. In simple terms, this allows root and non-root users to view older versions of files and revert changes. Allows you to configure schedule for the backups, automatically deletes old snapshots. The only sad thing is that Snapper has no updates since 2014.
backup2l is a lightweight command line tool for generating, maintaining and restoring backups on a mountable file system (e. g. hard disk). The main design goals are are low maintenance effort, efficiency, transparency and robustness. In a default installation, backups are created autonomously by a cron script. The script, that I've found on sourceforge and even used on couple of my projects 5 years ago. But it is not being updated since 2009.
FlyBack is software for system backup and restore, which offers similar functionality to the Mac OS X Leopard's Time Machine. Linux has almost all of the required technology already built in to recreate it. FlyBack is a snapshot-based backup tool based on rsync. It creates successive backup directories mirroring the files users want to backup, but hard-links unchanged files to the previous backup. Plenty of settings, mostly build for desktop computers, simple UI.
TimeVault monitors files for changes and takes snapshots after some user-specified delay. It is a simple front-end for making snapshots of a set of directories. Snapshots are a copy of a directory structure or file at a certain point in time. They use very little space for files which have not changed since the last snapshot was made, as they use hard links that point to existing backups. TimeVault makes all the work silently in background and is fully automated solution. Currently it gets no updates but it was a good solution when it just released.
Box Backup is an open source, completely automatic, on-line backup system. A backup daemon runs on systems to be backed up, and copies encrypted data to the server when it notices changes - so backups are continuous and up-to-date (although traditional snapshot backups are possible too). All backed up data is stored on the server in files on a filesystem - no tape, archive or other special devices are required.
BitCalm makes it easy for web developers to set up backup of applications on Linux servers just in one minute. It is SaaS for server backups. After installing python client user can manage backups for files and even databases in web-interface.