Good news for Ubuntu users (Hardy, Jaunty, Karmic and Lucid): Adobe Flash Player 10.1 final - which was released seconds ago - has already been uploaded to the Ubuntu partner repository.
To install Adobe Flash Player 10.1 final in Ubuntu, make sure you enable the Ubuntu Partner repository by going to System > Administration > Software Sources, on the second tab called "Other software" and enable the "http://archive.canonical.com/ubuntu CODENAME partner" repository (where CODENAME is hardy, jaunty, karmic or lucid).
Then do an upgrade and you should get the new Adobe Flash Player 10.1. In case you installed a different Flash package, install the one in the partner repository using the following command:
sudo apt-get update && sudo apt-get install adobe-flashplugin
Flash Player 10.1 is also available in Ubuntu 10.10 Maverick Meerkat multiverse repository.
Note: this is for 32bit only. There's no update for 64bit for now.