The average mattress contains millions of live and dead dust mites. These are tiny arachnids, belonging to the same family as spiders, which live on dead skin cells. They don't carry disease, and they don't bite. Most people have no problem with the inevitable contingent of mites in their homes. However, if you have an allergic reaction to mites, it might be necessary to take steps to control them. The main solution for a mattress is to enclose it in a plastic, mite-proof wrap and regularly clean other bedding, such as pillows, at high temperatures.
Bedbugs are tiny biting insects that have seen a resurgence in North America in recent years, probably due to increased international travel for business and leisure. The signs of bedbugs are itchy bites on waking, dots of blood on the sheets, and tiny black grains -- their droppings. Because bedbugs are elusive, especially during the day, you may not see the bugs themselves, as they'll be hidden in the mattress, bedding or surroundings. There is no guaranteed way to eradicate them from a mattress. You are probably best off starting with the range of non-toxic natural solutions, and if these don't work, calling a pest-control company.
Fleas, another group of insects, may also take up residence in bedding and mattresses. Fleas lay their eggs off the host animal, often in carpets and soft furnishings. If you have pets, those itchy morning bites may well be coming from fleas rather than bedbugs. Fortunately, fleas are rather easy to treat. A combination of treating pets, regular and thorough vacuuming of carpets and soft furnishings, and using any natural or commercial remedies you like is usually enough to treat the problem in time. Just remember to only use commercial flea treatments exactly as directed on the packaging. Using a pesticide for something other than its intended purpose, such as a pet flea powder on bedding, could be dangerous.
Depending where you live, you may come occasionally across other invertebrates on your bed, such as beetles or spiders. There aren't many bugs aside from dust mites, fleas and bedbugs that set up permanent homes in mattresses. Normally, flicking the bug into a cup and putting it outside, or killing it, is all you need to do. If you are unsure what you have found and whether it will cause problems, try an online identification forum.