Capillary waves are perhaps the simplest example to consider for an introduction to wave turbulence. Since the first paper by Zakharov and Filonenko [1], capillary wave turbulence has been the subject of many studies but a didactic derivation of the kinetic equation is still lacking. It is the objective of this paper to present such a derivation in absence of gravity and in the approximation of deep water. We use the Eulerian method and a Taylor expansion around the equilibrium elevation for the velocity potential to derive the kinetic equation. The use of directional polarities for three-wave interactions leads to a compact form for this equation which is fully compatible with previous work. The exact solutions are derived with the so-called Zakharov transformation applied to wavenumbers and the nature of these solutions is discussed. Experimental and numerical works done in recent decades are also reviewed.