Some predators work better than others for mosquito control
Mosquitoes have a variety of natural predators, from bats to fish to dragonflies. Scientists have studied what eats them in the wild, in hopes of finding enemies that can be introduced to get rid of mosquito problems near where people live. Some predators eat enough to noticeably reduce the population, while others get rid of so few that one can barely tell any difference.
Bats, birds and dragonflies which kill adult mosquitoes are some of the most obvious predators, but the least effective, according to the book Medical and Veterinary Entomology.  Mosquitoes make up only a small part of their diet. Fish, insects and other creatures that feed on larvae are another possible form of control, though still only effective in certain circumstances. One of the most effective natural enemies is Bacillus thuringiensis israelensis or Bti, a bacterium that only targets mosquito larvae plus a few very closely related species, without affecting fish, frogs or other desirable species, including people.
Two natural predators which attack mosquito larvae in the water are mosquitofish and dragonflies:
The gambusia fish (Gambusia affinis), also called the western mosquitofish, eats mosquito larvae. These little fish are controversial because they're aggressive, also eating the eggs, larvae and young of native fish and amphibians and competing with them for other food. They're not always successful at controlling mosquitoes, with unpredictable results. There is also a related species, Gambusia holbrooki, the eastern mosquitofish, which is sometimes used for mosquito control too.  They're tiny, grayish fish.
People have tried stocking gambusia to control mosquitoes in various areas of the United States since the early 1900s, with varying success. Due to the harm to native wildlife, mosquitofish are regulated in many states, where they may only be stocked in enclosed ponds or pools that have no outlet to streams or rivers, so the fish can't escape into the wild. A permit may be required.
The Rutgers Center for Vector Biology reported on several tests of mosquitofish in various situations. In an unused swimming pool, they stocked 35 fish each spring, since freezing of the water's surface caused them to die off in the winter. The fish ate all the larvae of Culex and Aedes mosquitoes, and also ate Anopheles mosquito larvae if there was no floating vegetation. If there was vegetation on the surface, such as might be found in an ornamental lily pond, the Anopheles larvae used it to hide and the gambusia fish couldn't control them.
The Virginia Department of Game and Inland Fisheries suggests the fish are "somewhat effective" in ponds, drainage ditches or containers, but also cautions they must not be able to escape through an outlet into other water. Gambusia may also overwhelm local native species, including dragonflies, killifishes or topminnows, which also control mosquitoes, so there may be no actual improvement in the problem, just a change from one kind of fish to another.
Gambusia are best used in isolated, artificial bodies of water with little to no surface vegetation. If you're interested in the possibility of using them for mosquito control in your area, contact your state's fish and game or wildlife department to find out restrictions and recommendations. In certain limited circumstances, they may work, but in others they may not work, or do more harm than good.
Other fish also eat mosquito larvae, so gambusia aren't the only choice. The popular and ornamental Koi fish are too large to feed on tiny mosquito larvae, but smaller guppies, killifish or common goldfish will live alongside Koi and eat them. This article discusses other fish which also eat mosquitoes.
The larvae can live in water too shallow for even small fish to reach them, so if you want to use fish to control them, make the sides of the pond steep, so there's little shallow water at the edges.
Dragonflies are natural enemies of mosquitoes in all their stages of growth. Dragonfly larvae live in the water like mosquito larvae and feed on them, while adult dragonflies eat adult mosquitoes.
The problem is that dragonflies also eat other things. When the mosquito population starts to get low, the dragonflies will choose other prey instead, leaving fewer mosquitoes than before, but not eliminating the problem.
Dragonflies also require a habitat similar to mosquitoes, so adding standing water to attract dragonflies will attract mosquitoes as well. If an area is already free of pools and puddles, better to leave it that way and try to deal with the mosquitoes that fly in from elsewhere in some other way. Fish which eat mosquito larvae will also eat dragonfly larvae, so both fish and dragonflies may live in balance in nature, but they'll compete with each other if you're trying to get them both to multiply so they can overwhelm the mosquitoes.
An experiment in Yangon, Myanmar, in Asia, showed that dragonfly larvae could be introduced to successfully control mosquitoes that were breeding in large containers of drinking water.  But like gambusia, dragonfly larvae work best under specific conditions, such as small artificial ponds or other containers of water where there are few other insects to eat besides mosquito larvae, and no vegetation for the mosquitoes to hide in, and no competing fish or other wildlife.
In the United States, pest control experts discourage purchasing dragonflies for mosquito control, since other alternatives are more effective in most circumstances. Purchased dragonflies will probably not be local species, and introducing non-native species may cause more problems to the local ecosystem.<
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 Medical and Veterinary Entomology, Gary Mullen, Gary Richard Mullen, Lance Durden.
 Gambusia holbrooki Fact Sheet and Gambusia affinis Fact Sheet
 Dragonflies & Damselflies, edited by Alex Cordoba-Aguilar.
Public domain photos of gambusia courtesy of the CDC. Dragonfly photo by Aleksandar Spasic, Stockxchng.