Hand luggage – check in advance with your airline for weight and size restrictions!

Here’s FSN’s list of recommended hand-luggage essentials for babies, toddlers and older kids:

For babies:
Nappies
Wipes
Sterilised bottles for feeding
Formula for the flight in case of delays (check with your airline about how much you can carry and in what form – liquid/powder).
Two changes of clothes for baby (including something warm) and an extra set of clothes for you (in case of spillage/sickness)
Medical and first aid supplies
Small, easily clutchable soft toys, dummies, rattles, etc
Antiseptic wipes
Disposable bibs
A variety of snacks (if your baby’s on solids).  Cabin crew will warm baby food and bottles upon request.
Ziplock bags for unused food, dirty nappies, etc
Tissues
Spare pacifier/blanket/muslin – in case one gets dirty or lost
A couple of empty plastic bags and/or sick bags

For toddlers, add:
Toys, books, puzzles, etc
Neck pillows for long-haul flights.  Kids are more likely to sleep if they’re comfortable (Most long-haul airlines provide blankets and some provide pillows too) and they can sit on them if they’re struggling to see the movie if the person sitting in front of them is tall!
Emergency tantrum kit – only to be used as a last resort to avoid a full-on tantrum (ie your child’s favourite lollipop/treat/toy – whatever you know will quieten them down if the going gets really tough!)

For older kids, simply add:
Electronic devices
(NB: Remember to remind your children to fully charge them!)