Last year Epic Games ported their Unreal Engine 3 to Adobe Flash Player 11 using Adobe's Stage 3D low-level API. Stage 3D exposes full GPU-hardware-accelerated 2D/3D APIs for Flash and AIR to web-browsers as well as iOS and Android.
Earlier this year at GDC2012, Epic Games then showed off UE3 running in an Adobe Flash browser in the form of their Epic Citadel demo, Dungeon Defenders, and Unreal Tournament 3. Details on the UE3 Flash port can be found at UnrealEngine.com.
Unfortunately, Adobe hasn't supported Stage 3D on Linux with hardware-accelerated rendering. As a result, while Unreal Engine 3 will run in Flash on other operating systems, due to the Adobe Flash Linux client not supporting the APIs with hardware support, it hasn't worked.
With Adobe abandoning Flash on Linux and issuing their last Flash Linux release in March, it looked like this Stage 3D support with being able to tap the GPU hardware would never come to Linux.
The good news though is that with Google taking over maintaining Flash for Linux within their Chrome web-browser, they have gone ahead and implemented the hardware support for Stage 3D that works on Linux. Stage 3D support is currently disabled by default in Chrome, but it can be easily toggled. After enabling the support within Chrome and ignoring any GPU black-listing, the Unreal Engine 3 demos work on Linux under Flash.
There's this Phoronix Forums thread where Unreal Engine 3 via Stage 3D is being discussed. It looks like a few bugs remain though with some texture problems and other issues while some aren't seeing any problems.
This isn't as good as seeing the native Linux port of Unreal Engine 3, which was completed by Ryan Gordon with a native Linux port of the Unreal Tournament 3 game, but it doesn't look like Unreal Tournament 3 will never be publicly released for Linux. At the moment it doesn't look like things will be different with Unreal Engine 4.
For those wanting to play the full Unreal Tournament 3 game on Linux outside of using Flash Stage 3D, the Windows version should work under Wine.