  We consider the problem of setting up the Wigner distribution for states of a
quantum system whose configuration space is a Lie group. The basic properties
of Wigner distributions in the familiar Cartesian case are systematically
generalised to accommodate new features which arise when the configuration
space changes from $n$-dimensional Euclidean space ${\cal R}^n$ to a Lie group
$G$. The notion of canonical momentum is carefully analysed, and the meanings
of marginal probability distributions and their recovery from the Wigner
distribution are clarified. For the case of compact $G$ an explicit definition
of the Wigner distribution is proposed, possessing all the required properties.
Geodesic curves in $G$ which help introduce a notion of the `mid point' of two
group elements play a central role in the construction.
