Until today, it was not possible to share or send files on Facebook. Some third party apps did allowed to send files on Facebook but Facebook did not support it. Now Facebook is rolling our a file sharing feature.
File sharing will be initially available only for Facebook Groups. Just like images and videos, group members will be able to send and share files on Facebook.
Users will be able to upload files of size up to 25 MB. And it will not be possible to share music files like .mp3 or .wav files and executable files like .exe files.
Facebook is still serious about copyright issues, so if a file is reported by other users, it will be removed. Copyright issues is probably the reason why file sharing and sending feature is not rolled out throughout the Facebook.
At the time of writing, File sharing should be available on most Facebook groups. If its not available on your group, then it will be available within a few days.