If your cats begin to eliminate outside their cat litter boxes, don't punish them...they aren't doing it out of spite. Find out whether this behavior is due to a medical or a behavioral reason. Some medical disorders that make urination painful...such as feline urinary tract disease (FLUTD) often cause cats to associate the litter boxes with pain, leading to box avoidance.
Cats with disorders that cause an increased volume of urine...such as diabetes in cats or kidney failure...may refuse to use the box because the box becomes too damp for their preferences. You can have your vet check by giving your cat a physical exam and analyzing samples of his blood and urine. If that's not the cause you will have to delve further and do some serious sleuthing.
If your cat has suddenly started soiling after a lifetime of perfect etiquette with the cat litter boxes, the behavior may be due to a traumatic event your cat experienced while in his box or to a recent anxiety-producing change in his environment.
But if your cat has just occasionally made mistakes in the past and is getting progressively worse, your investigation should proceed with a close inspection of the evidence.
Perhaps he's marking his turf (a whole subject by itself) or avoiding his litter box because the litter box has some characteristic that he's developed an aversion to. For instance, was the litter brand recently changed or a scented litter now replacing an unscented one that your cat got accustomed to.
They also have definite preferences about litter texture such as particle size and weight...probably some other features we don't know about...that affect the feel of the litter.
My cats prefer a small particle size and I think large particles may be uncomfortable for their little paws to dig into. But if the material is an appropriate size but either too light or too heavy, your cat still won't find the sensation appealing.
You may notice him shaking his paws when he gets out of the box, or quickly running out when he's finished "his business". The box itself may also cause your cat to want to "go" elsewhere. If you suspect he's developed an aversion to the current box and he's okay with the actual litter, offer him a new litter box and perhaps that will eliminate the unwanted behavior.
It's enough to boggle the mind when you see all the different sizes and shapes of cat litter boxes to choose from...even electronic ones. As you face this selection of boxes, remember, cats don't care about fancy shapes and stylish colors.
What matters to your cat is having a clean, comfortable place to do his "business". Fancy litter boxes are more for your sake than for your cat's. There is a good selection at Amazon.
Most cat litter boxes with modifications are an attempt to spare owners from the task of scooping. Some boxes have nested grid-work trays designed to sift an entire load of litter at once. Other designs sift the litter into a side compartment as the box is rolled upside down.
The most extravagant cat litter boxes are the electronic "self-cleaning" versions. It uses an electric eye to sense when your cat is finished...and 10 minutes later...a motorized rake sifts the waste into a plastic bag. All you need to do is dispose of the bag.
However, while many of these products sound good, they're not quite the labor-saving devices they're meant to be. Sifting trays still require owners to bend down and remove the waste. Also, boxes with built-in sifters are not as easy to keep clean. The edges and crevices are traps for wet litter, urine and bacteria.
The same is true of the electronic litter box, and it has another disadvantage; the motor and raking mechanism take up a lot of room in the box, so the actual litter surface area is rather small. Any cat that uses it must have a pretty good aim.
For a box your cat will gladly use, keep it simple. Pick a rectangular or square design with smooth curved sides, without edges and crevices. Smooth sides not only protect your cat from rough edges, they make cleaning easier.
Also, make sure the box is big enough. Some cats are reluctant to use small boxes and it increases the chances that your cat will "overshoot" the edge. Mainly, it has to be low enough for your cat to step over the edge and big enough for comfort.
For the aged or disabled cat, there is also a low profile litter box shown below.
As for litter box covers, they do shield the owner from the unpleasant smells and sights, but if the box looks and smells unpleasant to us, it's probably unpleasant for the cat as well. Instead of hiding what's in it...just get rid of it.
As long as the box is large enough, your cat may be perfectly content. Many covered cat litter boxes come with replaceable charcoal inserts to eliminate urine smells, but these thin filters can't absorb enough odor to make a difference.
Where odor is concerned, I find the most important thing is to clean the box frequently. Scoop it out at least once per day, and if you aren't using clumping litter, empty the entire box once a week or more often as needed. I find that I scoop the litter boxes 3 or 4 times a day because I have multiple cats. If your cat goes outside his box, and you've tried larger boxes and box locations, check with your vet to make sure it's not a medical problem.
Location is also important for the cat litter boxes. Keep the box away from eating areas. Cats are often reluctant to urinate or defecate in areas where they have their meals. Also, don't put the box in a heavily trafficked area.
Your cat may find it hard to relax, especially if there are other cats in the household that might stage an ambush. Finally, make sure the box is in a room where you will see it and remember to clean it regularly.