  In this paper, we study continuous Kakeya line and needle configurations, of
both the oriented and unoriented varieties, in connected Lie groups and some
associated homogenous spaces. These are the analogs of Kakeya line (needle)
sets (subsets of $\mathbb{R}^n$ where it is possible to turn a line
(respectively an interval of unit length) through all directions {\bf
continuously, without repeating a "direction"}.) We show under some general
assumptions that any such continuous Kakeya line configuration set in a
connected Lie group must contain an open neighborhood of the identity, and
hence must have positive Haar measure. In connected nilpotent Lie groups $G$,
the only subspace of $G$ that contains such an unoriented line configuration is
shown to be $G$ itself. Finally some similar questions in homogeneous spaces
are addressed.
  These questions were motivated by work of Z. Dvir in the finite field
setting.
