Legal Question in Family Law in California

You, unless your divorce judgment says otherwise.

Specifically, the dog is your separate property as you acquired it before marriage. Assuming she paid nothing to assist you in acquiring the dog, she has no claim whatsoever. Even though many emotionally think otherwise, a dog (or cat) would not be considered a "child" and, as such, would not be the proper subject of custody orders that swing into action with respect to children of divorced parents.