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This information is only aimed at residents of the European Union.
If you or your child are travelling within Europe, they just need a national identity card or passport.
In addition to their own valid passport or ID card, all children travelling alone, or with adults who are not their legal guardian, or with only one parent, may need an extra (official) document signed by their parents, second parent or legal guardian(s) authorising them to travel.
You should first consult the local embassy of the country the children are travelling to for information on which, if any, other documents they need to make the trip.
If you or your child are travelling outside of Europe, you will need a passport and sometimes a visa. Check with your transport organisation. Visas need to be obtained from the embassy of the country concerned. Some countries issue visas immediately, but for safety's sake, allow a month's lead time. For the USA you will probably need an ESTA authorization.