The Canon EF 20-35mm is a highly rated wide-angle lens that suffers from a common AF malfunction. Many users find the focus control stops working in both auto and manual mode. An excellent article by Piers Hendrie illustrates very clearly how to take apart the lens, and one possible fix for the issue.
The cause of this problem is that the USM ring motor (#4 below) is not applying enough pressure to the focus ring (5). Pressure is applied to the focus ring via a spacer (1), wave washer (2), and a felt ring(3). When you have this issue you can still hear the USM motor moving, but the focus ring does not move.
Piers believes the issue is that the felt washer has worn and it now too thin. He solves this by adding additional shims, which fixes the issue for him.
I think the more likely cause is that the wave washer has relaxed and is no longer applying enough pressure. I verified this by increasing the height of the spring (dimension C in the below figure.)
I did this by pinching each bend on either side with my thumb and forefinger and increasing the amount of bend in each of the waves of the washer. Doing this increases the height of the washer, and makes the diameter slightly smaller, so when you go to put it back on you kind of have to stretch it to get it on.
After reassembly the autofocus work great, and no additional shims were required.