  We analyze the behavior of causal geodesics on a Kerr-de Sitter spacetime
with particular emphasis on their completeness property. We set up an initial
value problem (IVP) whose solutions lead to a global understanding of causal
geodesics on these spacetime. Causal geodesics that avoid the rotation axis are
complete except the ones that hit the ring-like curvature singularity and those
that encounter the ring singularity are necessary equatorial ones. We also show
the existence of geodesics that cross or lie on the rotation axis. The
equations governing the latter family show the repulsive nature of the ring
singularity. The results of this work show, that as far as properties of causal
geodesics are concerned, Kerr-de Sitter spacetimes behave in a similar manner
as the family of Kerr spacetimes.
