There's been some 64-bit Linux demos of Unreal Engine 4 available previously, while now they're all centrally listed via the Linux Demos on the Epic Wiki. Demos available right now include Elemental, Elemental Cave, Realistic Rendering, Reflections Subway, Sci-Fi Hallway, Mobile Temple, Stylized, and Blueprint Examples. There's also the basic Tappy Chicken, Swing Ninja, and Card games offered too.
These games are only offered for 64-bit Linux as Epic is making the (right) choice of foregoing 32-bit support on Linux. Ubuntu Linux and SteamOS remain the primary distribution targets by Epic Games but these demos/games should run on other distributions too. More information on the current Unreal Engine 4 Linux support level can be found via the Unreal Engine Wiki.