  It is now widely accepted that the universe as we understand it is
accelerating in expansion and fits the de Sitter model rather well. As such, a
realistic assumption of black holes must place them on a de Sitter background
and not Minkowski as is typically done in General Relativity. The most
astrophysically relevant black hole is the uncharged, rotating Kerr solution, a
member of the more general Kerr-Newman metrics. A generalization of the
rotating Kerr black hole to a solution of the Einstein's equation with a
cosmological constant $\Lambda$ was discovered by Carter \cite{DWDW}. It is
typically referred to as the Kerr-de Sitter spacetime. Here, we discuss the
horizon structure of this spacetime and its dependence on $\Lambda$. We recall
that in a $\La>0$ universe, the term `extremal black hole' refers to a black
hole with angular momentum $J > M^2 $. We obtain explicit numerical results for
the black hole's maximal spin value and get a distribution of admissible Kerr
holes in the ($\Lambda$, spin) parameter space. We look at the conformal
structure of the extended spacetime and the embedding of the 3-geometry of the
spatial hypersurfaces. In analogy with Reissner-Nordstr\"{o}m -de Sitter
spacetime, in particular by considering the Kerr-de Sitter causal structure as
a distortion of the Reissner-Nordstr\"{o}m-de Sitter one, we show that spatial
sections of the extended spacetime are 3-spheres containing 2-dimensional
topologically spherical sections of the horizons of Kerr holes at the poles.
Depending on how a $t=$ constant 3-space is defined these holes may be seen as
black or white holes (four possible combinations).
