For the most part, cats are quite clean in what they do and how they take care of themselves. They spend a lot of time grooming themselves and usually tidy up before leaving the litter box. Sometimes something happens and a cat with stop covering up their mess or stop using the litter box altogether.
If your cat has completely stopped using the litter box you need to figure out why because this can become a nasty little problem. First, you will want to make sure your cat is in good health. Make sure your pet can get in and out of the litter box without help. Health problems could prevent him from being able to climb into the box. There is a possibility that your cat has decided he doesn't like where the litter box is located. There are cases where the cat feels alone to go to another part of the house to the litter box or there is too much activity in the area where the litter box is located. You could try moving the box to another area and see if that makes a difference. There is also the possibility that something happened while your cat was using the litter box. If he was scared or thought he was being attacked while in the box there may be some fear of going back.
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If the problem is more liquid than solid, your cat may be marking his territory. This is usually done but males that haven't been neutered but females that haven't been spayed can do it as well. Cats can mark their territory inside of a house if they see cats or other animals outside or if they feel crowded inside. Cats aren't overly social animals and if you have more than one cat they might feel crowded. Even bringing home the smell of another cat can set of the instinct to mark their territory.
One common problem is when a cat doesn't cover her mess in the litter box. This could be because of something simple like a new litter being used or the cat not thinking there is enough litter to use. Male cats will sometimes leave their business uncovered as a means of marking their territory. Try more litter in the box, then a different kitty litter. There is also the possibility that your cat has suddenly decided she doesn't like the type of litter box she has.
Another easy thing to check is to make sure the litter is clean. Cats, being the clean creatures they are, don't like to do their duties in a box that is already dirty. A litter box should be cleaned once per day and if you have multiple cats you might be better off to clean it more than once a day. Sometimes multiple cats cause a territorial battle with the litter box and one or more cats will decide they want their own box. Adding another litter box in a different area of the house is another option to try.
Problems with cats not using the litter box or not covering their mess are quite common. A little trial and error can usually solve most of these problems. If you try everything and it doesn't help consider seeing a veterinarian as there may be a hidden health concern bothering your pet.