You can sue your landlord for negligence.  Negligence is the failure to exercise due care (that degree of care that a reasonable landlord would have exercised under the same or similar circumstances to prevent foreseeable harm).Since your state requires all windows to have screens, the landlord's violation of that statute establishes negligence.Your damages (the amount of compensation you are seeking in your lawsuit) would be the vet bills for your cat's injuries.You may be able to file your lawsuit in Small Claims Court.  Your damages should also include the court filing fee and process server fee.