Cunard is the only cruise line that will allow you to transport your dog or cat on the cruise with you for a fee of around $1,000/$1,500 per pet. Cunard has a kennel area onboard the ship which includes a full-time kennel master. These sailings book up extremely early (12-18 months in advance) so be sure to reserve your cruise in advance.
All other cruise lines ban pets however exceptions are made for service animals. Requirements for traveling with your service dog vary by cruise line.