Here are the results of a study on veg*n dogs, which lists the pros and cons of the diet:
http://vegepets.info/diets/veg_canine.html
Dogs are not obligate carnivores, meaning they can be perfectly healthy on a nutritionally balanced vegan diet.
Some people have moral qualms about it not being 'natual' to feed a dog vegan food, but really, there is nothing natural about domesticated dogs, from their dependence on humans, to the standard surgical procedures and medical treatment they receive, to their pet food diets - even raw meat diets are not the same as eating whole animals, and can be contaminated with dangerous bacteria. Dogs are scavengers, meaning that naturally, they will eat whatever is available. In the wild that might be meat more often than not, but most dogs will happily eat the veggie dishes you serve them in your home. Taurine and L-carnitine may need to be supplemented if you are making their food at home, but any nutritionally complete vegan dog food should contain these already. You can provide some fresh/cooked veggies as well for optimal health, but be aware of which foods are unsuitable for dogs - some are listed here:
http://www.aspca.org/pet-care/poison-control/people-foods.aspx and also the seeds/pits of stone fruit, persimmons and apples contain cyanide and should be avoided.
It is also possible for cats to survive as vegans, but cats ARE strictly carnivores, so it takes a lot more work and strict monitoring of their health. I personally would never adopt a cat because I would not be comfortable serving meat OR withholding meat from a cat. But I've seen people on this forum who do go to those efforts and have perfectly healthy vegan cats.
Some vegan dog foods:
Veganpet -
http://www.veganpet.com.au/
All vegan Australian company, exceeds AAFCO's nutritional requirements for all stages of life, all ingredients are human grade, all natural except for some synthetic vitamins, and organic where possible. Has been tested on willing doggie participants for palatability and digestibility (people's pets, not lab animals.)
Vegepet -
http://www.vegepet.com.tw/
(Can be bought here:
http://www.vegepooch.com.au/index.php/catalogsearch/result/?q=vegepet)
Cruelty-free company in Taiwan, 'Functional' range is available in puppy, adult and senior/overweight, and 'Super Premium' range (suitable for dogs with corn/wheat/soy allergies) is available in adult and senior. All meet appropriate AAFCO requirements.
BiOpet -
http://www.biopetonline.com.au/productsVeganDog.html
Vegan option from non-vegan company, meets AAFCO nutritional requirements for adult dogs.
V-Dog -
http://v-dog.com/
Vegan company in the US, meets AAFCO requirements for adult dogs, hypoallergenic, gluten free, no GMOs. Unfortunately, they will only ship to Australia for orders of 1000lbs+ at this stage... I'm mentioning it because it is the only vegan dog food I've found that is completely free from corn, wheat and soy (even the Vegepet Super Premium contains hydrolysed soy & maize, although it is low allergen.) I hope an Australian store will stock V-Dog one day!
Veggie treats:
VegePooch -
http://www.vegepooch.com.au/index.php/products/dogtreat.html
Variety of brands, mostly vegan, some vegetarian.
Vegan Online -
http://www.veganonline.com.au/pets
Veganpet and Happier Pets brands, both vegan, can be purchased from their own websites as well.