Most shelters make every effort to screen animals for serious health and behavior problems.
Even with the best-behaved animal, however, you should expect to go through a period of adjustment as your pet becomes used to a new home, family, and routine. No animal, no matter where it comes from, is problem free or comes with a health and behavior guarantee, but if you're prepared to provide your new family member with some time, training, and patience, your reward will be a loving companion. If you are interested in a specific breed, ask if your local shelter keeps a waiting list of people interested in purebred animals. Even if it does not, you may be able to find an animal that has traits that are similar to those of the breed you have in mind. Many shelter animals are puppies and kittens that will be starting their lives as companions.