I use rsync on gnome-ubuntu 15.10 to back up my data to my server running Debian 8. This creates incremental backups similar to Apples Time Machine. The backup runs every 2 hours so this creates more backups than needed at the expense of hard drive space. I used to manually delete the files from the server and would try to save a monthly backup, 8 weekly backups, 30 daily backups, and 2 weeks of every 2 hour backups. This was a time consuming process of manually selecting the files and thus I was not consistent about removing the extra backups. My backup scripts are written in python and I was going to write a script that would delete old backups that were not needed any more. Even better than writing your own script is finding one that has already been written such as https://rotate-backups.readthedocs.org/en/latest/#rotate-backups-simple-command-line-interface-for-backup-rotation. This script will automatically delete your old backups and you can configure it for many backups you want to keep.
This script is well documented and easy to use. I give it my highest recommendation.