The majority of dogs put down in shelters are mutts; mixed breeds of unknown origin. All dogs are awesome, but there are many reasons to adopt a mutt. For instance, mixed breeds are thought to adapt more easily than purebreds.
Purebred dogs adhere to standard when it comes to both appearance and temperament. A mutt is unique and you get a one-of-a-kind. That also means they’re not as strongly “programmed” as purebreds, and tend to be more flexible.
If you want a hunting dog or a herding dog, you probably want a purebred that is developed to perform these specific tasks. If you want someone to keep you company around the house, a mixed breed can be much easier to handle than say, a hyper-energetic Border Collie that wants to herd anything that moves.
Many mixed breeds make great service and therapy dogs; they often combine the best properties of their parents. They are also generally healthier than purebreds, because they’re not as prone to specific illnesses.
If you’re the competitive kind, your mixed breed might shine. Many excel in agility, dock diving, flyball, and similar. The UKC even has a special class for mixed breeds.