  We study the set of connected components of certain unions of affine
Deligne-Lusztig varieties arising from the study of Shimura varieties. We
determine the set of connected components for basic $\s$-conjugacy classes. As
an application, we verify the Axioms in \cite{HR} for certain PEL type Shimura
varieties. We also show that for any nonbasic $\s$-conjugacy class in a
residually split group, the set of connected components is "controlled" by the
set of straight elements associated to the $\s$-conjugacy class, together with
the obstruction from the corresponding Levi subgroup. Combined with
\cite{Zhou}, this allows one to verify in the residually split case, the
description of the mod-$p$ isogeny classes on Shimura varieties conjectured by
Langland and Rapoport. Along the way, we determine the Picard group of the Witt
vector affine Grassmannian of \cite{BS} and \cite{Zhu} which is of independent
interest.
