Flash was instrumental in delivering audio, video and animations to the Internet in the late 90s but it has since evolved into one of the most frowned upon pieces of software in the modern era. Most mobile browsers no longer support Flash due to a number of issues including security, battery life and performance but it's still in heavy use on the desktop.
If the format isn't going to die anytime soon, it would seem that the next best step is to replace Flash Player. That's exactly what Mozilla has been doing since early 2012 with a project called Shumway. It uses HTML5 and JavaScript to offer a run-time processor for SWF and other rich-media formats. The good news is that it will work on platforms for which runtime implementation aren't available.
There's still a lot of work left but you can now try out the Flash Player alternative in the latest Firefox Nightly build (version 27). It isn't activated by default so you'll need to navigate to about:config and turn it on. Interestingly enough, you still need to have the Flash Player installed (for now).
Shumway isn't yet capable of running all commercial Flash apps but some early demos on Mozilla's online Shumway Inspector look promising. Whether or not the project will ultimately be able to replicate all of Flash Player's capabilities remains to be seen.
The final build is expected to trickle down into a public Firefox release early next year.