It seems that Adobe is getting less and less cozy with Linux, despite having built 64 bit support for Flash initially only on Linux. Adobe recently announced that they are no longer releasing Flash Player for Linux other than bundled with Google Chrome. Support for the current version of Flash Player, version 11.2, will continue for 5 years in the form of security updates.
This is yet another slap in the face for Linux users, with Adobe previously withdrawing the Linux version of Adobe AIR, which many open source developers latched onto when it was first unveiled.
Having said all of that, it is worthwhile to know that Adobe Flash Player is not the only option out there. For instance, using an up-to-date browser, you can join the HTML5 beta programme on YouTube and watch videos natively in your browser. I've been testing this out, and it works pretty well. It is slightly buggy at times with fullscreen video not always working properly, and sometimes the video can be choppy, based on how busy your computer is at the time and how well videos normally work.
Another alternative is the open source Gnash Player. This project was started many years ago as an open source implementation of Flash Player, but received very little positive press. To their credit, the development team continued undeterred and apparently Gnash can now even play most YouTube videos in addition to most Flash-based sites. I haven't tested it out myself, but I'm strongly considering doing so.
One more alternative is another open source Flash replacement called Lightspark. It is both a standalone player, as well as a brower plugin, just like Gnash is.
None of these options covers each and every instance of Flash, but they will most likely work in almost all situations. Additionally, things seem to be moving more and more toward open standards, and so it may not be too long before things like Flash and Silverlight become a thing of the past.