Make a system backup.
Methods to backup and restore the Eee PC are found on the Tips And Tricks page.
Determine your model.
Please find the Model page for your exact model number and read it first. Asus releases new models at a dizzying pace, often with deceptively similar-looking model numbers that have entirely different hardware in them, so it is important to know what you have to choose an appropriate install method.
Select an appropriate install method.
If you want to install Debian Squeeze (recommended), Use the Standard Installer HowTo. This should cover most users, as you can install from the network or without, as well as special needs like pppoe, lvm or crypto.
If you want to install to a USB key or SD/SDHC card, follow SD Card or USB HowTo.
If you have no USB key, external hard drive, SD/SDHC card or CD/DVD-ROM, for some models it’s possible to boot the standard installer over the network (though you currently need a USB ethernet adaptor). See Netboot HowTo.
If you happen to want to install Debian Lenny (deprecated) on certain old Eee PCs, follow the Lenny On Old Models HowTo; the debian-eeepc.img will install a proper Debian system and provides a few necessary drivers and acpi-scripts to properly support some of the Eee PC hardware. The installer supports braille displays and speakup should work too (this one needs testing).