How long before my baby can play with my dog?

I live in Sweden currently and I've seen families who have no qualms about having their pets interact with their babies even at a young age. The dog or cat seem to know that the baby is a combination of its owners and welcome the baby happily. Obviously these animals are house pets that have been cleared of diseases etc. As long as the interaction is supervised by an adult, it should be fine. So many cute videos about pets and babies being best friends :)