So, you've been knitting your fingers off on a lovely sweater, and the top is great, but despite your best attempts at self-deception, you're finally forced to face it: it's just too long. Hits at a horrible spot, bags around your hips, completely not working. It would be perfect if you could just take a couple of inches off; unfortunately, you knitted it from the bottom up with a ribbed waistband, and ripping out ribbing from the cast-on row is a real bear. Or maybe you just hate the bottom ribbing, and wish you'd done a straight or rolled hem. Or maybe it's even too short, and you'd like to have an extra couple of inches above where the ribbing is.
The answer is sweater surgery. In most cases, it's possible to cut off the offending bottom, and reknit the bottom of the sweater in a manner more to your liking.
You're going to need a thin circular, preferably several sizes smaller than the size you knitted the sweater with; the length isn't terribly important as long as it's enough to go comfortably around the sweater (or across the front or back half, if it was knitted flat). If you don't have a suitable needle, you can use a tapestry needle threaded with a smooth yarn or a piece of thin ribbon, but you'll need to transfer the stitches back to a needle before you can knit them, whereas with a thin circular you can knit from it directly. You'll also need some good sharp scissors.
Turn the sweater upside down, so the area to be removed is at the top; if it was knit flat and has already been seamed, you'll want to undo the seams in the part to be removed so you can work with the edge stitches. Select a row to become your live stitches, from which you will knit your new sweater bottom. Using the thin circ, pick up the right leg of each of these stitches. Just to be extremely clear here: sometimes when people say to "pick up" stitches, they mean to pull a loop of the working yarn through each stitch, forming a new stitch on the needle -- that is not what I mean. You want to actually capture the right leg of the existing stitches on the needle; no working yarn is involved at all. If you get the left leg instead, it isn't a big deal, but you'll need to knit the stitches through the back when you start knitting down, so they're not twisted. Just check them as you work that first row, and be sure you're knitting through the right leg of every stitch, whether it's in front or in back.
Once you've captured all the stitches for your new working row, it's time to cut the excess part of the knitting free. You want to leave the first several stitches on the row above the captured row intact; these will provide the tail for you to weave in, where you join your working yarn to knit downward. Leave about two horizontal inches worth of stitches for a 6" tail, a few more if you prefer a longer tail to work with.
Now, there are a couple of ways to do the cutting, depending on how bold you are. You can snip a single stitch, choosing the stitch right after the tail stitches you're leaving intact, and then unravel one row stitch by stitch. You can cut straight across the knitting a couple of rows above the needle you placed, and then unravel back down to the needle, which may require picking out some stitches one by one, depending on the type of stitch. Or, you can cut each of the stitches on the row right above the needle, in which case it's important to remember that for that first couple of inches, you want to cut two rows above, so the tail stitches stay intact; this last option leaves you the least amount of subsequent work to do, but it takes a little more nerve. Whichever option you choose, once you've separated the stitches on one row, the former sweater bottom will come off in a piece, and can then be easily unraveled top-down to reclaim that yarn, if you wish.
You're now ready to join yarn and knit the new bottom of your sweater, but there are a few things to note before you begin. First, because you're now working with what were formerly the bottoms of your stitches, you're offset by half a stitch. This isn't a problem if you're working in stockinette; because you're also going in the other direction, the new stitches will nest nicely into the old ones, and the point where you changed direction will be no more noticeable than a well-done grafting job, which is to say generally completely invisible. However, you'll also have one stitch fewer, which means you may need to increase by one, or decrease by one fewer if you previously increased after the ribbing, in order to make your ribs come out evenly.
If you're working in something other than stockinette and you're not going straight into a new bottom ribbing, the half-stitch offset becomes an issue, because the jog will show. The solution is to change your stitch pattern -- do a band of something else for half an inch or an inch, and then switch back to your pattern stitch, and that will fool the eye sufficiently. If your design won't work with something like that, then you probably don't want to try this technique.
Are there other situations where this technique shouldn't be used? Yes, of course; nothing's universal. In particular, you'll notice that in the opening paragraph, every situation I mentioned involved a ribbed bottom. There's a reason for that. Ribbing is a major hassle to rip from the cast-on edge; the way you move the yarn back and forth results in the yarn being woven through the stitches as they appear from the bottom up, so you have to unpick every individual stitch, and that makes cutting the ribbing off in a piece really appealing. Stockinette, on the other hand, rips as easily from the bottom as from the top, so for a rolled-hem sweater, it's better to unpick the cast-on row and rip upward than it is to cut.
This technique can, of course, be used on sleeves and other situations, not just sweater bottoms. While it won't rescue every knitting misstep, it can be a real lifesaver in the situations it's suited to.
ETA: I've just recently become aware of this nice little tutorial which shows this method with pictures. Enjoy!