We discuss the construction of the holonomy groupoid associated with a
singular foliation as a quotient space. This groupoid is highly
pathological, nevertheless we explain that our construction allows for the
generalization of the $C^*$-algebra of the holonomy groupoid to the
singular case. We also discuss how this $C^*$-algebra is the recipient for
the index of elliptic longitudinal pseudodifferential operators, as well
as quantization problems.