Here's an example of why you don't want to:
Barbaric conditions in the airline industry, by a shelter mom.
If you are forced to, if it is inescapable that you must travel with you animal companion, one of the very important things to remember is don't assume any thing. Many airplanes these days will not accept animal companions. The ones that do, accept only certain types. Also, if you are able to have your companion ride with you in the passenger cabin, you must make sure that your carryons fit that plane. All airplanes are not the same. Check with your specific the airline for regulations on animal companions and dimensions of the carryons for transport especially cabin transport.
International travel is a whole other nightmare.
Very, VERY risky.
First off, most airlines do not allow animals in the passenger cabin. They get thrown in the hold with the baggage. On each and every airline site, you will find a statement like this:
"Pets As Checked Baggage"
Do you really want to take that chance seeing as how your luggage is treated??
Another major problem is that frequently the information and policies that you need about the receiving country (the country you're flying into), are not available or WORSE, not up to date in your home country. I ran into that problem with Japan. Not only could I not get the info I needed from the US side, their official Japanese site didn't have it either.
I played it safe and turned out to be right in not trying to transport them with me.
Yet another problem with international travel is the QUARANTINE procedures--there's papers you need from licensed vets and a quarantine stay that requires additional fees. It can be a quite a lengthy stay, too.
Sites Specializing in Traveling with Animal Companions:
Pets On The Go: lots of information. Basic but MINIMAL fee to join. Well worth the cost.
Animal Airlines: company specifically geared to transporting your loved ones safely.
Direct Links to the Most Common American Airlines:
Thanks again for stopping into my little corner of the 'net, and Happy Browsing!!