  We carry out a careful study of basic topological and ergodic features of
Delone dynamical systems. We then investigate the associated topological
groupoids and in particular their representations on certain direct integrals
with non constant fibres. Via non-commutative-integration theory these
representations give rise to von Neumann algebras of random operators. Features
of these algebras and operators are discussed. Restricting our attention to a
certain subalgebra of tight binding operators, we then discuss a Shubin trace
formula.
