Litter box problems
Cats are exceedingly clean animals and the most common reason for not using the litter is that it simply isn't clean enough for them. We humans wouldn't use a toilet that was messy so we shouldn't expect our cats to do the same. Added to that is the fact that their sense of smell is far superior to ours - it may smell clean to us but it may not to a cat.
The addition of another cat to the house making use of the same box can also put them off - the smell of the new cat may make them think that the litter is the other cat's territory and not for them to use. Remember in the wild, urine is used to mark territory and while pet cats don't have the same concerns as wild cats, the urges and instincts are still there.
Changing either the litter box or the litter used in it can stop cats making use of it. This is particularly common if you have changed style of box, such as from an open box to a closed one. The new box may seem intimidating or dark to them and they won't use it so make alternative arrangements elsewhere in the house. Similarly, the litter inside may not feel comfortable on their paws or if it is scented, they might not like the smell. All of these and other problems may stop them using their litter box.
Not using the litter box properly can also be a sign of a health problem and may need a trip to the vet. Cats that have some type of urinary tract infection may come to associate the litter box with the pain of trying to urinate and therefore go elsewhere.
A cat with a kidney, liver or thyroid problem may also get 'caught short' and be unable to get to the litter box in time to use it. This is because they may need to use the box more frequently due to the condition and the urge comes on quicker than their ability to get to the box. Similarly, a cat with diarrhoea or constipation may not have the time to get where they need to before depositing their waste.
Observing the problem
Watching the cat is key to figuring out the problem. For example, if they cry out when using the litter tray, claw at something or choose to use some fresh dirt rather than their dirty litter box, there are all indicators of one of the problems we have mentioned. One test is to confine the cat to a room with a clean litter box - if he chooses not to use it and deposits waste elsewhere then either the litter box or the litter may be the problem.
If you have any inclination that your cat has a medical problem, then the best course of action is to see your vet. They can check out the cat thoroughly and even if the problem is behavioural rather than physical, advise you on a course of action.