Just complementing @kos's nice answer:
The Zero day exploit (CVE-2016-0728) was found by the researchers at Perception Point.
It was introduced in 3.8 Vanilla Kernel. So the Ubuntu releases using kernel versions 3.8 or later are vulnerable. As @kos already mentioned, on a supported release you can just upgrade the kernel to install the patch:
sudo apt-get update && sudo apt-get dist-upgrade
How to Check the Vulenerability (taken from here):
sudo apt-get install libkeyutils-dev
Copy this C program from github, written by the Perception Point team to test the exploit.
Save the codes in a file e.g.
test_2016_0728.c and compile:
gcc test_2016_0728.c -o test_2016_0728 -lkeyutils -Wall
Run the compiled program as normal user:
It would take some time, if your kernel is vulnerable after the completion of the program you will see that your privilege has been escalated i.e. you have become