  In this paper we discuss some exact results related to the fractional
Klein--Gordon equation involving fractional powers of the D'Alembert operator.
By means of a space-time transformation, we reduce the fractional Klein--Gordon
equation to a fractional hyper-Bessel-type equation. We find an exact analytic
solution by using the McBride theory of fractional powers of hyper-Bessel
operators. A discussion of these results within the framework of linear
dispersive wave equations is provided. We also present exact solutions of the
fractional Klein-Gordon equation in the higher dimensional cases. Finally, we
suggest a method of finding travelling wave solutions of the nonlinear
fractional Klein-Gordon equation with power law nonlinearities.
