Transporting an animal can be stressful for both the cat and the owner. Proper advance planning will minimize this for all involved. Some airlines allow pets to travel in-cabin with you, so check around.
Steps
Research international requirements at least three months prior to your departure. Many countries require animals to be inoculated two months prior to departure. If you're moving your pet internationally, you need to have an identification chip implanted.
Investigate the container requirements for whichever airline you choose. Generally,you will be safest purchasing a lockable hard sided plastic container with holes on all four sides. It must also be large enough for your pet to be able to stand up and turn. In-cabin carriers must fit under the seat, and may count as part of your carry on limit.
Get the animal shipping containers well before your flight date. This gives the cat time to acclimatize to the container. One way of speeding the acclimatization period is to feed your cat inside the cage or basket. Also, choose a container large enough to hold a small litter box during transit. An in-cabin carrier will be smaller, so should always be lined with an absorbent and waterproof pad. Disposable diapers are a good substitute
Attach your easily visible contact information to the container securely, so that it will not be accidentally pulled off.
Ensure that your pet has passed its health inspection within one month prior to departure. Some countries require that your pet visit a national government inspection center as well. You will need to find this out for the specific country that you're going to or coming from.
Check with the local TSA supervisor to fully understand security screening procedures at your departure airport, and at other airports you plan to visit. Be prepared to remove the animal from the container during screening.
Check to see what your airlines regulations are regarding shipping pets. You may need to change airlines to find one that accommodates pets. Be prepared for this.
Reconfirm with your airline 48 hours before departure that you will be bringing animals.
Reduce your cat's food to roughly half the usual the day before transport. However, ensure that it has an adequate water supply during the entire transit.
Arrive at the airport three to four hours early in case there are any last-minute problems.
Realize that most airlines charge to transport pets, and the charge is usually identical to the cost of an additional piece of luggage. Try to either pay this beforehand, or be prepared to pay on the day of your flight.
Cover your feline container with a light blanket. Something to calm the cat from sights and sounds and such, but which allows air to get to the cat.
Tips
A leash and harness are a must!! When passing through checkpoints you will often need to remove the animal, so have the harness on and the lead ready during the check in process. Help your pet get use to the harness by starting several weeks ahead of time.
Include several of your cat's favorite toys in the transport container - very few cats will say 'no' to catnip.
Remember to place a small litter box with high sides in the transit container.
Check with local cat breeders who typically show cats on the show circuit. The breeders should know the best airlines to use for flying your feline.
Warnings
Do not sedate your cat. The decreased air pressure in the airplane will magnify the effect of the drugs on your feline. This could harm or kill your cat.
Even the calmest pet can be frightened by strange sights smells and sounds - so keep your animal confined or leashed at all times. You do not want to be chasing a frightened animal around a strange airport in the middle of a distant city. An ounce of prevention is better than a pound of cure.
