Flash Player 10 adds new filters, 3D effects, better typography support, and more. For Linux users this version offers windowless mode support (transparency through Flash, and page elements rendered above Flash), V4L2 camera support, smooth fullscreen video playback, and better performance.
[update] Adobe still does not provide a 64-bit Flash Player. I don’t have a 64-bit Ubuntu installation to test on, but Alejandro left a link in the comments that could help 64-bit users.
The Flash Player in Ubuntu 8.10 has not been kept up to date with recent testing versions, so it’s not looking likely that this final release will be included now. However, Adobe is now making it much easier for Linux users to install the Flash Player.
[update] This version of the Flash Player is now available in Ubuntu 8.10! Install it from the flashplugin-nonfree (click link to install) package.

I’ve tested these steps on both Ubuntu 8.04 and 8.10.

1. You’ll need to first remove any previous versions of Flash Player you have installed. This includes manual installations and the flashplugin-nonfree package from Ubuntu. Use this command to remove Ubuntu’s Flash package:
   

   sudo apt-get remove flashplugin-nonfree

2. Visit Adobe’s download page, select the .deb for Ubuntu 8.04+ download, and install the package.
3. Restart your web browser.

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