This question, and most answers, are more than 10 years old now, and the situation has changed in the meantime.
Firstly, as has been said, don't put film in checked luggage. The scanners used for checked luggage are strong enough to damage film after one pass.
Putting film in your hand luggage used to be relatively safe. The effect is cumulative, and it used to be the case that film needed many passes through the scanner before there was any noticeable effect. The French civil aviation authority did some tests and found, for example, that ISO 400 negative film could withstand up to and beyond 24 passes through a hand luggage X-ray scanner (ref). However, airports are currently in the process of replacing old X-ray scanners with newer CT scanners. I know that all UK airports for example are supposed to have CT scanners fully rolled out by 2024. These newer scanners will harm film even at one pass. To be honest, it's not easy being a film photographer air passenger anymore. I would advise arriving at the airport with plenty of time to spare and requesting a hand inspection of your film, rather than allowing it through the scanner. Sometimes, it's necessary to be a little insistent, or to ask to see a supervisor. I was made to wait recently in Hamburg airport until a police officer was available to look at my camera, and he insisted that I open the camera, so of course I had to rewind mid-roll first. Most staff have no idea that these CT scanners can damage film, and some I think don't even know what film is! Kodak Alaris has made available some sticker templates that you can have printed to put on your film pouch, which may help to convince airport staff not to put your film through the scanner. Fujifilm too issued a short press release warning against these new scanners. I'm kind of hoping that there will be enough film photographers going through airports that the granting of hand inspection of film will become widely routine.
One other thing I do is always include a sacrificial roll of Ilford Delta 3200 with my film. I don't know how many times this roll has been through airport screening, and I never intend on actually shooting the roll! But sometimes airport staff have been told that film up to ISO 800 is fine to go through the scanner, and by having this one roll of 3200 with me, it sometimes gets me a little more sympathy.
The alternative to all of this is to buy film at your destination and have it developed before you return. I did that in Paris this year - but it's not so easy/cheap to do for many destinations. X-ray / CT scanners do no harm at all to film that has already been developed and fixed.