Children require passports for international travel (read my post on applying for your child’s first UK passport), but for flights within the UK the adult travelling with them can vouch for their identity. The adult will need to carry photo ID, the more official the better (though I was intrigued to learn that the airline Flybe includes NUS cards and valid firearm certificates on its long list of acceptable forms of identification).
When it comes to visas to almost any destination you care to name, you can safely assume that the rules are the same for children as they are for adults. This has implications not just for pre-trip admin, but for budgeting too – fees are typically the same regardless of the age of the applicant (though they do often vary depending on the nationality of the person applying, something to watch out for if your child holds a different passport to you).
None of the above probably comes as much of a surprise – we’re all used to needing passports and visas to travel. What you might not be aware of is that if you’re taking a child abroad, you technically need permission from anyone else with parental responsibility to do so – ie if you’re a mother or father travelling with your baby by yourself, you need to bring a letter specifying that their other parent has given the trip the go ahead. To really do it by the book, the letter should be witnessed by a notary, and you should bring along proof of your relationship to the child, such as a birth certificate – the real thing, not a photocopy. Mumsnet helpfully provide this template consent letter for travelling with children to make sure you don’t leave anything out.
There are very few situations in which you would actually be asked to provide such a letter, but some countries are stricter than others so it’s worth checking in advance. Parents who don’t share a last name with their children also report more hassle in this regard (the law is designed to prevent child abduction). Sorting out the permission letter – particularly having it notarised – certainly sounds like a pain, but much less of a pain than being refused entry at the border and being sent home.
Update: a commenter on Mumsnet told me that she always travels with her children’s birth certificates because they have a different last name to her, and that she is usually asked to show them at the UK border. I think I’ll put a copy of the baby girl’s birth certificate in the travel document wallet the next time we go away, just in case.