Damage can come from webbing clothes moths, casemaking clothes moths, and carpet beetles. This fact sheet from Colorado State University has great photos for identification. Here are some other helpful articles (and the one from the other day just to have this all in the same place).
There are several ways to get rid of moths. The fact sheets have similar methods but differ on temperatures and lengths of time. (Note: I didn't investigate pesticides or traps. You may want to look into these methods if you have a moth problem.) Here is what I'm doing:
I'm putting my wool (and other animal) yarn and fleece into Ziploc freezer bags and squeezing out any extra air. These bags are going into my chest freezer which is set at or below 0 degrees Fahrenheit. The bags will stay for at least 72 hours. Since freezing doesn't affects the eggs, the the bags will be brought to room temperature and remain there for at least 21 days to allow any eggs to hatch. The yarn will then be frozen again for 72 hours.
I'm going to heat my cotton yarn in the oven for 4 hours at 120 degrees F. Moth larvae won't eat the cotton yarn but they may lay their eggs there.
I'm washing all of my fabric in hot water and drying it on high heat. Again, the moths aren't in it but could be laying their eggs there.
Any fabric that is too sensitive to heat or too large to fit in the washer will be taken out in the sunshine and brushed off. (Moths hate sunlight.) I also did this with my loom bag. I turned it inside out and brushed it all out.
As I moved things from my sewing room, I washed the baseboards and vacuumed under them, paying special attention to the corners. All fabric and yarn was removed from the dark closet and will be stored in the sunnier room.
My wool was already in plastic bins but apparently that didn't help or the yarn went in after it was already infested. My yarn will now be stored in sealed Ziploc bags with extra air removed.
The curtains will be open more often in my sewing room to keep it nice and bright in there. The floors will be swept more often and the furniture will periodically be moved away from the walls for vacuuming at the baseboards. Any new yarn will be inspected and stored in its own plastic bag.
And I will be sorting through my stash to remove yarn I don't use. I've found a group that needs yarn and embroidery floss. After I'm sure any moths are removed from the yarn, I'm going to donate a bunch to them.
Then I get to move on to the master bedroom because my husband says he saw a small moth flying around in there a couple of months ago! Hands off my knitted shawls and sweaters, moths!!!