The most known Linux PDAs in these days are the Agenda VR3 by AgendaComputing (out-of-production), the iPAQ by HP/COMPAQ, the Zaurus series by SHARP, and the Yopy by Samsung (out-of-production). Except the iPAQ all of them are true Linux PDAs, they are pre-equipped with Linux by their manufacturers.
There are different free distributions for Linux PDAs available, e.g.: QT Embedded (pre-installed on the SHARP Zaurus), Opie, Familiar. The Gnome Palmtop Environment - GPE aims to provide a Free Software GUI environment for palmtop/handheld computers running the GNU/Linux operating system. GPE uses the X Window System, and the GTK+ widget toolkit.
Most of the software for the newer PDAs can be obtained as pre-compiled IPK packages. You may search the Zaurus Software Index - ZSI or ipkgfind for the package you need. To install these packages you may choose different methods. One method is to install directly via a HTTP connection called feed. For an example see the TuxMobil IPK feed.
Besides these well-known Linux PDAs I will also try to point to ports for other PDAs and to tools to achieve connectivity to non-Linux PDAs, cell phones and desktop computers.