  Magnetohydrodynamic (MHD) waves are often reported in the solar atmosphere
and usually classified as slow, fast, or Alfv\'en. The possibility that these
waves have mixed properties is often ignored. The goal of this work is to study
and determine the nature of MHD kink waves. This is done by calculating the
frequency, the damping rate and the eigenfunctions of MHD kink waves for three
widely different MHD waves cases: a compressible pressure-less plasma, an
incompressible plasma and a compressible plasma with non-zero plasma pressure
which allows for MHD radiation. In all three cases the frequency and the
damping rate are for practical purposes the same as they differ at most by
terms proportional to $(k_z R)^2$. In the magnetic flux tube the kink waves are
in all three cases, to a high degree of accuracy incompressible waves with
negligible pressure perturbations and with mainly horizontal motions. The main
restoring force of kink waves in the magnetised flux tube is the magnetic
tension force. The total pressure gradient force cannot be neglected except
when the frequency of the kink wave is equal or slightly differs from the local
Alfv\'{e}n frequency, i.e. in the resonant layer. Kink waves are very robust
and do not care about the details of the MHD wave environment. The adjective
fast is not the correct adjective to characterise kink waves. If an adjective
is to be used it should be Alfv\'{e}nic. However, it is better to realize that
kink waves have mixed properties and cannot be put in one single box.
