Of course it is advisable for children to use a restraining system adapted to their size and weight for them to travel more safely, but it is not an obligation.
Normally the airlines allow very small children, up to two years old, to travel on the knee of an adult using a seatbelt fixed to that of the adult, supplied by the crew (they usually travel free).
Generally, from two years of age, they can use the airplanes adjustable two point seatbelt.
Of course it would be better to use a more specific restraining system. If a parent wishes to do so, the airlines don’t usually object to you taking the child seat that you would normally use in the car on board with you, but is always a good idea to check with the airline that the seat is suitable for the type of aircraft you will be travelling on.
Obviously your child would be occupying its own seat for which you will be charged. Keep in mind that the seat, whether forward or backward facing, must be of a size compatible with the airplane seat and can be anchored to it correctly. The space in some planes is limited.
Generally speaking, the majority of child seats approved for use in a car will be suitable for use in an airplane. Recently, seats have started to carry a label, which states that they are “certified for use in motor vehicles and aircraft”. There shouldn’t be any problem in using one.
The airlines also have a list of which seats are compatible with which aircraft.
As well as child car seats, there is a type of seat designed specifically for airplanes, which has been certified in both Europe and the United States. It is often referred to as CARES.
It comprises of a four-point harness designed for use with the aircraft seat. It is fixed at the bottom by seatbelt and at the top by a strap supported by the back of the seat. It is not obligatory for the airline to carry them, but parents can purchase one to take with them. It is light (not even a kilo in weight) and takes up little space (fits in a handbag or backpack).