My only box has only 512 MB of RAM, a 1.15 GHz single-core 32-bit processor, and 2 hard disks (roughly 50 GB total), and it runs the 32-bit (i686) version of Arch Linux. I use mainly text-mode/console applications, usually in a terminal emulator in X. I use only a few graphical applications now (e.g., Firefox, The GIMP). I also do not use a desktop environment any more — I use xmonad, a tiling window manager.
Firefox is too large, though. It uses too much RAM, swapping often, and it takes as long as 10 seconds to load. I am considering switching to a smaller Web browser such as uzbl. I already sometimes use ELinks, a text browser, when I do not need something as large as Firefox.