Gather proof that you're responsible. The more documentation you can provide attesting to your conscientiousness as a pet owner, the more convincing your appeal will be to your future landlord. Written proof that your adult dog has completed a training class, or that your puppy is enrolled in one. A letter from your veterinarian stating that you have been diligent in your pet's medical care. Supply documentation that your pet has been spayed or neutered and vaccinated against rabies.
Most veterinarians routinely fulfill such requests for their clients.
Make your request to the individual or group with the ultimate authority to grant your request. Usually this will be the owner of the house or apartment. The owner may, however, delegate the decision to a property manager or resident manager. If you encounter a no-pets policy, ask if it is the result of a negative experience with a previous resident. Addressing your landlord's prior experience may show you how to present your own request most effectively.