From Fedora Project Wiki

Flash on Fedora 10

Suitable flash plugins (players) should be installed for playing flash content in web pages. Fedora provides an open source player (Gnash) by default. As of now it can deal with about 81% of the open Flash specifications. For more details see the [Gnash Manual].

The proprietary flash plugins provided by Adobe do not ship by default in Fedora 10. Others intending to use the proprietary flash players provided by Adobe may note that Flash Version 9 is NOT Supported. But Flash 10 is known to work in both the X86 and X86_64 architectures (with some hiccups).

X86_64 users need to install all of flash-plugin, nspluginwraper.X86_64 and alsa-plugins-pulseaudio.i386 for the 32-bit flash player. The last of these must be changed for the 64-bit alpha version.

X86 users need to install all of flash-plugin, nspluginwrapper.i386 and alsa-plugins-pulseaudio.i386' from adobe's yum repository. For SELinux related problems see this.