If a while back Adobe wasn't very interested in its 64bit Linux users and discontinued Adobe Flash Player 10.1 for 64bit, the new Adobe Flash Player 10.2 codename "Square" developer preview released by Adobe today is also available for 64bit Linux systems (and obviously, 32bit too).
To install Adobe Flash Player 10.2 "Square" developer preview in Linux (Ubuntu), download it, extract the archive and place the "libflashplayer.so" file into "/usr/lib/adobe-flashplugin/" (make a backup of the old file first) - you'll have to run Nautilus as root for this, so press ALT + F2 and enter:
gksu nautilus /usr/lib/adobe-flashplugin/
If that folder does not exist, see also: /usr/lib/flashplugin-installer/ or /usr/lib/flashplugin-nonfree/ (the installation folder depends on how you've installed Adobe Flash Player).
Have you tried the new Flash Player? How is it working for you?
Download Adobe Flash Player 10.2 "Square" developer preview (Available for Linux, Windows and Mac OSX - both 32bit and 64bit)
Update: the new Adobe Flash Player "Square" (10.2.x) 64bit is available in a PPA for Ubuntu 9.10, 10.04 and 10.10