I've got an Eee Box B202 now, using it for my main desktop with Ubuntu 8.04.1.
It runs well, though sometimes a tad slow - Firefox does occasionally get sluggish.
Ubuntu installed directly from an external DVD drive via USB without hassle, with a monitor connected via DVI.
There are two display problems:
I have the box mounted behind the monitor with the supplied VESA mount, and it's nice and tidy.