Can I sharpen a knife with sandpaper?
Yes, I've heard this suggested, using wet/dry sandpaper and a mousepad. It is a very inexpensive way to match a whetstone, and you can use sandpaper with the same grit to produce an excellent edge.

You duct-tape the sand-paper together so it wraps around the mouse pad, and then pull the knife along the sandpaper with the edge trailing. This is to say, you use sandpaper in the opposite direction as a whetstone.  Once you work up to the finest-grit papers, you can use jeweler's polish and a leather strop for a truly razor-sharp edge. 

Because the mouse pad's rubber/foam has some give, this method produces a convex edge which remains sharp for longer that an V-cut, because it has more metal behind the edge.  It also still presents a very sharp point for cutting, moreso than an equivalent V-cut front bevel.

Using sandpaper will also cost a small fraction of what you spend on a good sharpening stone, which will run you $50 or more PER STONE.

Wondering what grit to use?  There are comprehensive tables here and here.  However, here is a quick-reference mini-table for you:

Name / US Grit Rating / Use


Coarse / 100-400 / Remove lots of metal.  At the low-end of the range, used for reshaping blades, taking out large chips, or restoring snapped points.  At the higher end of grit range, used to establish a bevel angle and restore very dull edges.
Medium / 400-700 / Remove moderate amounts of metal. For refining an edge established with coarse grits, and for restoring moderately dull blades.  Can be used to establish a bevel angle, but it will take considerably longer.  600 grit is a common point to start at for standard sharpening.
Fine / 800-1200 / For finishing a blade or touching-up a slightly dull blade.  At about 1000 grit, you will get something equivalent to a factory edge, assuming you use the same bevel angle. Ceramic steels have grits in this range.
Extra-Fine / 2000+ / For polishing.  Higher grits create a shinier and sharper edge, until at 8000+ grit you get mirror finishes and sharpness equivalent to a straight razor.  If you're going to bother using grits this high, a leather strop and polishing paste may serve you better than even a ceramic steel; the steel would actually be reducing the edge sharpness with use.


Source: Sandpaper-mousepad sharpening

Source2: Sharpening techniques and explanation

Edit: Another way to use sandpaper in sharpening -- expanding the grit range for a Spyderco Sharpmaker

I'm currently using sandpaper to make a very coarse stone for sharpening extremely dull edges on my Spyderco Sharpmaker.  I wrap both of the normal triangular stones with a strip of emery cloth or 220-grit wet/dry sandpaper, and then hold it in place with medium binder clips.  Since the lowest stone the Sharpmaker comes with is 800-grit ceramic, this saves HOURS versus using the normal stones. It is also considerably cheaper than the diamond stones they sell; for a big tub of clips and a package of sandpaper, I paid $7, versus about  $37 for the diamond rods.