I just brought my 7 month old Scottish Fold kitten with me from Singapore 4 days ago. After a number of near-heart attacks after being told he wouldn't be shipped, I will share some information I learned.
Warn your airline IN ADVANCE with export and import (if needed) paperwork included! I told mine a month before but provided official paperwork a week in advance and that caused me much grief. Apparently, they need to contact each foreign port of arrival that the cat touches (in my case, Australia and New Zealand) and get the paperwork for each...it takes 'weeks' but if forced, they can do it in 2 days. You can't do it yourself, and I tried. You also have to confirm with your airport (not airline) cargo because they have additional requirements. I ran from one side of the airport to another getting signatures and rang up $70 in taxi fees, just at the airport. Remember that you'll need to be there many hours in advance. I almost wish I'd used a company, but they quoted me $1,000 more than I just spent.
As a warning, you will get smacked with random costs all over the place. I paid about $150 in fees when the cat landed. On top of a $500 airfare charge. Export permit cost. Approval fees. And 4 vet visits for check-ups and tests which will vary country by country. Luckily, my little guy didn't need quarantine because Singapore is so stringent with their animals.
Finally, don't bother to put food or toys or litter in the carrier. They will throw it all away. It was a shame because my boyfriend made the air travel container as livable as possible with all the cat amenities and it was all gone when the kitten arrived, bug-eyed and jet-lagged. He made it, though...and he's currently sitting next to me on the bed in the hotel, staring at this screen and cuddling. Totally worth it!