  We introduce a class of locally compact Hausdorff groupoids and show how to
associate C*-algebras to them in a way which generalizes the reduced C*-algebra
of an 'etale groupoid. Focusing on criteria for simplicity and existence of
Cartan subalgebras, we obtain results which both generalize and improve on the
corresponding results from the 'etale case. In the second part we apply the
results to dynamical systems and subshifts.
