Why are LiPo batteries so popular in the RC world?

What are LiPo batteries? And why are they so popular in the RC world?
LiPo batteries (short for Lithium Polymer batteries) are a type of rechargeable battery that has taken the electric RC world by storm, especially for planes, helicopters, and multi-rotor. They are the main reason electric flight is now a very viable option over fuel powered models.
RC LiPo batteries have four main things going for them that make them the perfect battery choice for RC planes and even more so for RC helicopters over conventional rechargeable battery types such as NiCad, or NiMH.
1) Light weight. RC LiPo batteries are light weight and can be made in almost any shape and size.
2) Large capacities. RC LiPo batteries have large capacities, meaning they hold lots of energy in a small package.
3) High discharge rates. RC LiPo batteries have high discharge rates to power the most demanding electric motors.
4) No "memory-effect". Unlike NiCad or NiMh; LiPo's have no "memory-effect".
In short, LiPo’s provide high energy storage to weight ratios, are capable of fast discharges, and come in an endless variety of shapes and sizes.
These benefits are important in any RC model, but for airplanes, helicopters, and multi-rotor they are the reason electric flight has become so popular.