Pet adoption, be it a cat, dog, hamster, or horse, is a very special thing to do. You are saving a life, helping an animal in need, and you will be rewarded with (hopefully) many years of joy. Adopting is slightly more tricky than buying, as adopters need to be approved. This is a guide on pet adoption.
* Please Note: Adoption is from proper animal shelters, it is NOT the same thing as buying from a pet store, breeder, or taking a pet out of the newspaper. Some people use the term "adoption" incorrectly."
Call the shelter a few days before you want to adopt your pet. Ask them when their adoption hours are. If adoption hours close at 5:00 pm you will want to be there at least 1 hour earlier to have a good look at the available pets. More time is needed if you are adopting a horse or dog, than if you are adopting a rabbit, for example.
If you are looking for a specific, unusual, type of animal, such as a Chinchilla, you may want to ask if they have any for adoption. Otherwise, if looking for a cat or dog, you might ask if there is a particular day of the week that is best to come. Saturdays are usually fairly busy so might not be the best days of the week to adopt a pet.
Ask the shelter what you need to bring. Generally they want picture identification, proof you are over 18 years of age. If you rent they also need landlord permission to have a pet, so either bring your contract, or the landlords phone number. They may want to contact your veterinarian, if you have one.
You may wish to inquire what their adoption fee is and what forms of payment they take. Adopting a pet is not free. Most shelters are non-profit and poorly funded. They have to pay for the vaccinations, deworming, and so forth on the pets. Your adoption fee helps cover this and is usually lower than if you were to take a pet to the vet and have this done on your own.

http://www.flickr.com/photos/lynnefeatherstone/442029059/ A cat in a shelter in London, England.
If you are adopting a cat, or small dog, bringing a carrier, or box is a good idea too. If you are adopting a large dog you will want to bring a leash and collar. If you are adopting a smaller animal you need to bring something to take it home in if it does not come with its cage.
When you arrive at the shelter talk to somebody and let them know what kind of animal you are looking for. Be specific, do you want a cat that is good with dogs, a dog that is good with kids, or a rabbit that comes with its own cage? What age, gender, etc. of pet are you looking for? This will help the staff point you in the right direction.

http://www.flickr.com/photos/lilyinnepal/4062994851/ A dog for adoption in Singapore.
When you find a pet you might like ask the staff if you can take it out of its cage, or in the case of dogs, take it for a walk.
If you own a dog and are adopting a dog you may want to bring your pet to make sure the two will get along.
When you make up your mind ask the staff for an adoption questionnaire, or whatever they do to complete the adoption process. You may want to have them hold the animal for 24 hours while you get supplies (do not buy food before you buy your pet, you should stay on the same kind of food the shelter is feeding for a few days and change it gradually later).
If you do not find a pet to your liking ask if they have a waiting list. This is great when looking for a specific type of pet or breed. Ideally the shelter will call you when a pet comes in.
Keep in touch with the shelter after your adoption. Most are there to help you if you have any questions or problems later on. They will also want to know when you spay or neuter the pet if that is a requirement of the adoption.
Remember most shelters have contracts that if at any time you have a problem and cannot keep the pet, they will take it back.
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