Do I need a consent letter to travel with my child (UK)?

Travelling internationally with the correct paperwork can help ensure your trip runs without a hitch. But before children can travel abroad with one parent, they may need to fulfil certain requirements set by immigration officials of that country, which can vary. In some cases, verbal consent may be sufficient, but border officials often ask for proof of parental permission to prevent child abduction. So, can consent be withheld by the other parent? What needs to be in the letter, and what format must it take?