All three of my dogs were rescues. First was from an actual rescue organisation, a pup of about 4 months, terrier cross of some sort. Had him till 17 and he was brilliant all the way through (until near the end, when he became incontinent, but that's forgivable in such an old boy) even with small boistrous toddlers.
Second dog was a bitch a friend found running loose on a rural bit of main road. It was before the days when everyone had the internet, so she'd notified the relevant authorities but hadn't taken her to a shelter because of the risk of euthanasia. Her dh wouldn't let her add the dog to their existing menagerie, so we took her on. She was about 18 months old, and lived til she was about 16/17. She was another mongrel, with beagley looks and colouration, accustomed to indoor living and housetrained already. Great with everyone, and had the most amazing howl - she sounded like the Hound of the Baskervilles, though she was only about 15" at the shoulder! 
Third was a Border Collie who was advertised as "free to good home" on a card in the vets when we went to pick up the ashes of dog no 1. She'd been bred for working, but failed sheepdog training, and had just turned 3 yo when we took her. She settled amazingly well into family life, having never been in a house - she was kennelled outdoors before coming to us. I don't remember housetraining her specifically, she just seemed to know! Again, she was lovely with everyone (apart from cats, who terrified her) all the way from babies to old ladies, and died last year at 17.
Each of these dogs had their own characters and personalities, each had minor irritations about them, each was very much loved.
(And they were all very healthy till very old age... )