  We construct the holonomy groupoid of any singular foliation. In the regular
case this groupoid coincides with the usual holonomy groupoid of Winkelnkemper
(1983); the same holds in the singular cases of Bigonnet and Pradines (1985)
and Debord (2001), which from our point of view can be thought of as being
"almost regular". In the general case, the holonomy groupoid can be quite an
ill behaved geometric object. On the other hand it often has a nice
longitudinal smooth structure. Nonetheless, we use this groupoid to generalize
to the singular case Connes' construction of the C*-algebra of the foliation.
We also outline the construction of a longitudinal pseudo-differential
calculus; the analytic index of a longitudinally elliptic operator takes place
in the K-theory of our C*-algebra.
  In our construction, the key notion is that of a bi-submersion which plays
the role of a local Lie groupoid defining the foliation. Our groupoid is the
quotient of germs of these bi-submersions with respect to an appropriate
equivalence relation.
