Would you allow a dog or cat around your infant?

I see a lot of videos of dogs and cats being the perfect fairy godmother to their new infant babies, but what's the reality? I've heard dogs can be too excitable to be around an infant, and cats can become jealous of the infant and start destructive behavior.

What do you think? Many people who have babies also have a dog or cat in their life, how do you make everybody happy when the baby comes?