Read and rate the best solutions below by giving them a "thumbs up".
If you use cheese often, like I do, here is a decent tip! Cut straight through a block of unopened cheese, remove the portion you will use without demolishing package. Keep the wrapper from the piece you used, turn it over and cut a slice in the back side. Now, slip it over the remaining section of cheese. You now have a free wrapping technique, so no need to use up another dish or plastic bag/wrap. It works great for me, without drying out!
By Becca from NY
Keep cheese from going moldy by soaking a cloth in white vinegar and then wrap the cheese in the cloth. Put the cloth and cheese in an airtight container and refrigerate.
By Robyn 
To store partial blocks of cheese, which were originally purchased in a sealed plastic package: Wrap the remaining cheese, first in old fashioned waxed paper, then in foil. The waxed paper will draw moisture away from the cheese and it will last much longer in your refrigerator. As a side note, to shred very soft cheeses, such as mozzarella, put them in the freezer for about 10-15 minutes first. This will make the cheese firm enough to grate without it melting in your hands.
Give a "thumbs up" to the solution that worked the best! Do you have a better solution? Click here to share it!
Here are questions related to Storing Cheese.
I stored some cut up cheese chunks in a vacuum sealed bag for a week. There were droplets of water inside the bag in a few places (minimal). Do you suggest I place a paper towel in with the cheese chunks to prevent mold from building up due to excessive humidity? How much paper?
Thanks, (...and thanks to Harlean for her help with other queries.)
Holly from Richardson, TX
By Connie (Guest Post)11/01/2006
I keep my cheese in tupper ware (tight tops) and then either place a couple of pieces of cube sugar or a cap (about catsup bottle size) packed with brown sugar. replace when soft.
I have no trouble with mold with this method.