With the increased popularity of adoption, more animal lovers than ever are turning to shelters to find that special cat or dog. But for some pets, the journey to a "forever home" is much more challenging than others-with physical or behavioral needs that require extra love and care.

Take Goose, for example: an elder pup with a distinctive "honk" and a severe heart condition, who still grasped at life with joy and gusto. Or Tiny, who stayed at the shelter for months while the staff and veterinarians tried to figure out his condition, and was finally adopted by a couple who loved...