DVDs did not play out of the box on my Ubuntu Desktop 16.04 LTS machine. I was able to fix this by doing the following.
First, entered the command:
$ sudo apt install libdvd-pkg
After the package and dependencies finished downloading and installing, it launched a short guided prompt sequence. I answered yes to all prompts — the only one really worth paying attention to was a choice to have the thing automatically update itself. It sounded like it has to build itself from source every time or something, so I guess it is a little non-standard in this regard. The prompts then told me to run the following command, which I did:
$ sudo dpkg-reconfigure libdvd-pkg
This popped up another little pseudo-graphical terminal prompt screen which, if I recall correctly, just asked me if I wanted to install the thing, which I answered in the affirmative.
After I completed these steps, all was well and I can now play the two DVDs I have tested without any problems.
I discovered this fix via a page on the Ubuntu wiki, RestrictedFormats/PlayingDVDs. According to the wiki, the actual software package that does the work is called
libdvdcss2, and the
libdvd-pkg package is a convenience package that makes the install easier.